{"id":557,"date":"2019-03-07T13:24:10","date_gmt":"2019-03-07T13:24:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/unit-circle-sine-and-cosine-functions\/"},"modified":"2020-02-28T15:10:24","modified_gmt":"2020-02-28T15:10:24","slug":"unit-circle-sine-and-cosine-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/unit-circle-sine-and-cosine-functions\/","title":{"raw":"3.3 Unit Circle: Sine and Cosine","rendered":"3.3 Unit Circle: Sine and Cosine"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Find function values for the sine and cosine of [latex]30^{\\circ}\\text{ or }\\left(\\frac{\\pi }{6}\\right),\\text{ }45^{\\circ}\\text{ or }\\left(\\frac{\\pi }{4}\\right)[\/latex] and [latex]{60^{\\circ}}\\text{or}\\left(\\frac{\\pi }{3}\\right).[\/latex]<\/li>\r\n \t<li>Identify the domain and range of sine and cosine functions.<\/li>\r\n \t<li>Use reference angles to evaluate trigonometric functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Figure_05_02_001\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"370\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132215\/CNX_Precalc_Figure_05_02_001.jpg\" alt=\"Photo of a ferris wheel.\" width=\"370\" height=\"246\" \/> <strong>Figure 1:<\/strong> The Singapore Flyer is the world\u2019s tallest Ferris wheel. (credit: \u201cVibin JK\u201d\/Flickr)[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137692173\">Looking for a thrill? Then consider a ride on the Singapore Flyer, the world\u2019s tallest Ferris wheel. Located in Singapore, the Ferris wheel soars to a height of 541 feet\u2014a little more than a tenth of a mile! Described as an observation wheel, riders enjoy spectacular views as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs.<\/p>\r\n\r\n<div id=\"fs-id1165137714296\" class=\"bc-section section\">\r\n<h3>Finding Function Values for the Sine and Cosine<\/h3>\r\n<p id=\"fs-id1165134153045\">To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1, as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_002\">Figure 2<\/a>. The angle (in radians) that [latex]t[\/latex] intercepts forms an arc of length [latex]s.[\/latex] Using the formula [latex]s=rt,[\/latex] and knowing that [latex]r=1,[\/latex] we see that for a <span class=\"no-emphasis\">unit circle<\/span>, [latex]s=t.[\/latex]<\/p>\r\n<p id=\"fs-id1165137428004\">Recall that the <em>x- <\/em>and <em>y-<\/em>axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV.<\/p>\r\n<p id=\"fs-id1165137837083\">For any angle [latex]t,[\/latex] we can label the intersection of the terminal side and the unit circle by its coordinates, [latex]\\left(x,y\\right).[\/latex] \u00a0 Consider an angle that is in the first quadrant.\u00a0 We can drop a perpendicular line to the x-axis to create a right triangle. The sides of the right triangle will be x and y. \u00a0 If we use our right trigonometric definitions from Section 3.1, we can see that [latex]\\mathrm{cos}\\left(t\\right)=\\frac{x}{1}[\/latex] and [latex]\\mathrm{sin}\\left(t\\right)=\\frac{y}{1}.[\/latex]\u00a0 This means [latex]\\left(x,\\text{ }y\\right)\\text{ }=\\text{ }\\left(\\mathrm{cos}\\left(t\\right),\\text{ }\\mathrm{sin}\\left(t\\right)\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_05_02_002\" class=\"small\">[caption id=\"\" align=\"aligncenter\" width=\"371\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132218\/CNX_Precalc_Figure_05_02_002.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and an arc created by the angle with length s. The terminal side of the angle intersects the circle at the point (x,y).\" width=\"371\" height=\"293\" \/> <strong>Figure 2:<\/strong> Unit circle where the central angle is [latex]t[\/latex] radians[\/caption]<\/div>\r\n<div id=\"fs-id1165137638636\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165135560820\">A <strong>unit circle<\/strong> has a center at [latex]\\left(0,0\\right)[\/latex] and radius [latex]1[\/latex] . In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle [latex]t\\text{.}[\/latex]<\/p>\r\n<p id=\"fs-id1165137714560\">Let [latex]\\left(x,y\\right)[\/latex] be the endpoint on the unit circle of an arc of arc length [latex]t.[\/latex] The [latex]\\left(x,y\\right)[\/latex] coordinates of this point can be described as functions of the angle [latex]t[\/latex] where:<\/p>\r\n<p style=\"text-align: center\">\u00a0 [latex]\\mathrm{cos}\\left(t\\right)=x[\/latex] and<\/p>\r\n<p style=\"text-align: center\">[latex]\\mathrm{sin}\\left(t\\right)=y.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137726457\" class=\"bc-section section\">\r\n\r\nNote that this definition allows us to use angles which are not acute or in other words, angles in standard position whose terminal side is not in the first quadrant.\r\n<p id=\"fs-id1165137768253\">Now that we have our unit circle labeled, we can learn how the [latex]\\left(x,y\\right)[\/latex] coordinates relate to the <span class=\"no-emphasis\">arc length <\/span>and <span class=\"no-emphasis\">angle<\/span>. The sine function relates a real number [latex]t[\/latex] to the <em>y<\/em>-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle [latex]t[\/latex] equals the <em>y<\/em>-value of the endpoint on the unit circle of an arc of length [latex]t.[\/latex] In <a class=\"autogenerated-content\" href=\"#Figure_05_02_002\">Figure 3<\/a>, the sine is equal to [latex]y.[\/latex] Like all functions, the sine function has an input and an output. Its input is the measure of the angle; its output is the <em>y<\/em>-coordinate of the corresponding point on the unit circle.<\/p>\r\n<p id=\"fs-id1165137748691\">The cosine function of an angle [latex]t[\/latex] equals the <em>x<\/em>-value of the endpoint on the unit circle of an arc of length [latex]t.[\/latex] In <a class=\"autogenerated-content\" href=\"#Figure_05_02_003\">Figure 3<\/a>, the cosine is equal to [latex]x.[\/latex]<\/p>\r\n\r\n<div id=\"Figure_05_02_003\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"371\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132221\/CNX_Precalc_Figure_05_02_003.jpg\" alt=\"Illustration of an angle t, with terminal side length equal to 1, and an arc created by angle with length t. The terminal side of the angle intersects the circle at the point (x,y), which is equivalent to (cos t, sin t).\" width=\"371\" height=\"172\" \/> <strong>Figure 3:\u00a0<\/strong> Illustration of an angle t, with terminal side length equal to 1, and an arc created by angle with length t.[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137463311\"><strong>Important Note:\u00a0\u00a0<\/strong>Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses: [latex]\\mathrm{sin}\\text{ }t[\/latex] is the same as [latex]\\mathrm{sin}\\left(t\\right)[\/latex] and [latex]\\mathrm{cos}\\text{ }t[\/latex] is the same as [latex]\\mathrm{cos}\\left(t\\right).[\/latex] Likewise, [latex]{\\mathrm{cos}}^{2}\\left(t\\right)[\/latex] is a commonly used shorthand notation for [latex]{\\left(\\mathrm{cos}\\left(t\\right)\\right)}^{2}.[\/latex] Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer.<\/p>\r\n\r\n<div id=\"fs-id1165135251444\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165131968080\"><strong>Given a point <em>P\u00a0<\/em>[latex]\\left(x,y\\right)[\/latex] on the unit circle corresponding to an angle of [latex]t,[\/latex] find the sine and cosine.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165134108592\" type=\"1\">\r\n \t<li>The sine of [latex]t[\/latex] is equal to the <em>y<\/em>-coordinate of point [latex]P:\\mathrm{sin}\\left(t\\right)=y.[\/latex]<\/li>\r\n \t<li>The cosine of [latex]t[\/latex] is equal to the <em>x<\/em>-coordinate of point [latex]P: \\text{cos}\\left(t\\right)=x.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_02_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137417674\">\r\n<div id=\"fs-id1165137824250\">\r\n<h3>Example 1:\u00a0 Finding Function Values for Sine and Cosine<\/h3>\r\n<p id=\"fs-id1165137463869\">Point [latex]P[\/latex] is a point on the unit circle corresponding to an angle of [latex]t,[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_004\">Figure 4<\/a>. Find [latex]\\mathrm{cos}\\left(t\\right)[\/latex] and [latex]\\mathrm{sin}\\left(t\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_05_02_004\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"371\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132224\/CNX_Precalc_Figure_05_02_004.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (1\/2, square root of 3 over 2).\" width=\"371\" height=\"293\" \/> <strong>Figure 4:\u00a0<\/strong> Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the given point[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137436467\">[reveal-answer q=\"fs-id1165137436467\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137436467\"]\r\n<p id=\"fs-id1165135417749\">We know that [latex]\\mathrm{cos}\\left(t\\right)[\/latex] is the <em>x<\/em>-coordinate of the corresponding point on the unit circle and [latex]\\mathrm{sin}\\left(t\\right)[\/latex] is the <em>y<\/em>-coordinate of the corresponding point on the unit circle. So:<\/p>\r\n\r\n<div id=\"eip-id1165135678435\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}x&amp;=\\mathrm{cos}\\left(t\\right)=\\frac{1}{2}\\\\ y&amp;=\\mathrm{sin}\\left(t\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2} \\end{align*}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137723202\" class=\"precalculus tryit\">\r\n<h3>Try It #1<\/h3>\r\n<div id=\"ti_05_02_01\">\r\n<div id=\"fs-id1165137767886\">\r\n<p id=\"fs-id1165137767887\">A certain angle [latex]t[\/latex] corresponds to a point on the unit circle at [latex]\\left(-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}\\right)[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_005\">Figure 5<\/a>. Find [latex]\\mathrm{cos}\\left(t\\right)[\/latex] and [latex]\\mathrm{sin}\\left(t\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_05_02_005\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"370\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132227\/CNX_Precalc_Figure_05_02_005.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (negative square root of 2 over 2, square root of 2 over 2).\" width=\"370\" height=\"291\" \/> <strong>Figure 5:\u00a0<\/strong> Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the given point.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137604785\">[reveal-answer q=\"fs-id1165137604785\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137604785\"]\r\n<p id=\"fs-id1165137438734\" style=\"text-align: center\">[latex]\\mathrm{cos}\\left(t\\right)=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\text{ }\\mathrm{sin}\\left(t\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135596524\" class=\"bc-section section\">\r\n<h4>Finding Sines and Cosines of Angles on an Axis<\/h4>\r\n<p id=\"fs-id1165135438852\">For <strong>quadrantral angles<\/strong>, the corresponding point on the unit circle falls on the <em>x- <\/em>or <em>y<\/em>-axis. In that case, we can easily calculate cosine and sine from the values of [latex]x[\/latex] and [latex]y.[\/latex]<\/p>\r\n\r\n<div id=\"Example_05_02_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135173908\">\r\n<div id=\"fs-id1165137584111\">\r\n<h3>Example 2:\u00a0 Calculating Sines and Cosines Along an Axis<\/h3>\r\n<p id=\"fs-id1165137548354\">Find [latex]\\mathrm{cos}\\left(90^{\\circ}\\right)[\/latex] and [latex]\\text{sin}\\left(90^{\\circ}\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137418560\">[reveal-answer q=\"fs-id1165137418560\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137418560\"]\r\n<p id=\"fs-id1165137460268\">Moving [latex]90^{\\circ}[\/latex] counterclockwise around the unit circle from the positive <em>x<\/em>-axis brings us to the top of the circle, where the [latex]\\left(x,y\\right)[\/latex] coordinates are (0, 1), as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_006\">Figure 6<\/a>.<\/p>\r\n\r\n<div id=\"Figure_05_02_006\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"370\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132230\/CNX_Precalc_Figure_05_02_006.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (0,1).\" width=\"370\" height=\"291\" \/> <strong>Figure 6:\u00a0<\/strong>\u00a0Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (0, 1).[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137399724\">Using our definitions of cosine and sine,<\/p>\r\n\r\n<div id=\"eip-id1165135209632\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}x&amp;=\\mathrm{cos}\\left(t\\right)=\\mathrm{cos}\\left(90^{\\circ}\\right)=0\\\\ y&amp;=\\mathrm{sin}\\left(t\\right)=\\mathrm{sin}\\left(90^{\\circ}\\right)=1\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"eip-id1165134388912\" style=\"text-align: center\">The cosine of 90\u00b0 is 0; the sine of 90\u00b0 is 1.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137405943\" class=\"precalculus tryit\">\r\n<h3>Try It #2<\/h3>\r\n<div id=\"ti_05_02_02\">\r\n<div id=\"fs-id1165134302493\">\r\n<p id=\"fs-id1165137386643\">Find cosine and sine of the angle [latex]\\pi .[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137675196\">[reveal-answer q=\"fs-id1165137675196\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137675196\"]\r\n<p id=\"fs-id1165134205964\">[latex]\\mathrm{cos}\\left(\\pi \\right)=-1,[\/latex]\u00a0 \u00a0[latex]\\mathrm{sin}\\left(\\pi \\right)=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137864121\" class=\"bc-section section\">\r\n<h4>The Pythagorean Identity<\/h4>\r\n<p id=\"fs-id1165134198719\">Now that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is [latex]{x}^{2}+{y}^{2}=1.[\/latex] Because [latex]x=\\mathrm{cos}\\left(t\\right)[\/latex] and [latex]y=\\mathrm{sin}\\left(t\\right),[\/latex] we can substitute for [latex]x[\/latex] and [latex]y[\/latex] to get [latex]\\mathrm{cos}^{2}\\left(t\\right)+\\mathrm{sin}^{2}\\left(t\\right)=1.[\/latex]\u00a0 This equation, [latex]\\mathrm{cos}^{2}\\left(t\\right)+\\mathrm{sin}^{2}\\left(t\\right)=1,[\/latex] is known as the <strong style=\"font-size: 1rem;text-align: initial\">Pythagorean Identity<\/strong><span style=\"font-size: 1rem;text-align: initial\">. See <\/span><a class=\"autogenerated-content\" style=\"font-size: 1rem;text-align: initial\" href=\"#Figure_05_02_007\">Figure 7<\/a><span style=\"font-size: 1rem;text-align: initial\">.<\/span><\/p>\r\n\r\n<div id=\"Figure_05_02_007\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"371\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132232\/CNX_Precalc_Figure_05_02_007.jpg\" alt=\"Graph of an angle t, with a point (x,y) on the unit circle. And equation showing the equivalence of 1, x^2 + y^2, and cos^2 t + sin^2 t.\" width=\"371\" height=\"160\" \/> <strong>Figure 7<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137925355\">We can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or vice versa. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution.<\/p>\r\n\r\n<div id=\"fs-id1165137925357\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165137453137\">The <strong>Pythagorean Identity<\/strong> states that, for any real number [latex]t,[\/latex]<\/p>\r\n\r\n<div id=\"Equation_05_02_03\" style=\"text-align: center\">[latex]{\\mathrm{cos}}^{2}\\left(t\\right)+{\\mathrm{sin}}^{2}\\left(t\\right)=1.[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137731366\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137934414\"><strong>Given the sine of some angle [latex]t[\/latex] and its quadrant location, find the cosine of [latex]t.[\/latex] <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137901198\" type=\"1\">\r\n \t<li>Substitute the known value of [latex]\\mathrm{sin}\\left(t\\right)[\/latex] into the Pythagorean Identity.<\/li>\r\n \t<li>Solve for [latex]\\mathrm{cos}\\left(t\\right).[\/latex]<\/li>\r\n \t<li>Choose the solution with the appropriate sign for the <em>x<\/em>-values in the quadrant where [latex]t[\/latex] is located.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_02_03\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137724522\">\r\n<div id=\"fs-id1165135251259\">\r\n<h3>Example 3:\u00a0 Finding a Cosine from a Sine or a Sine from a Cosine<\/h3>\r\n<p id=\"fs-id1165137434023\">If [latex]\\mathrm{sin}\\left(t\\right)=\\frac{3}{7}[\/latex] and [latex]t[\/latex] is in the second quadrant, find [latex]\\mathrm{cos}\\left(t\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137812666\">[reveal-answer q=\"fs-id1165137812666\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137812666\"]\r\n<p id=\"fs-id1165137405175\">If we drop a vertical line from the point on the unit circle corresponding to [latex]t,[\/latex] we create a right triangle, from which we can see that the Pythagorean Identity is simply one case of the Pythagorean Theorem. See <a class=\"autogenerated-content\" href=\"#Figure_05_02_008\">Figure 8<\/a>.<\/p>\r\n\r\n<div id=\"Figure_05_02_008\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"370\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132235\/CNX_Precalc_Figure_05_02_008.jpg\" alt=\"Graph of a unit circle with an angle that intersects the circle at a point with the y-coordinate equal to 3\/7.\" width=\"370\" height=\"291\" \/> <strong>Figure 8:\u00a0<\/strong> Graph of a unit circle with an angle that intersects the circle at a point with the y-coordinate equal to 3\/7.[\/caption]\r\n\r\n<\/div>\r\n<p id=\"eip-id1165137727966\">Substituting the known value for sine into the Pythagorean Identity,<\/p>\r\n\r\n<div id=\"fs-id1165134393100\" class=\"unnumbered\">\r\n\r\n[latex]\\begin{align*}\\mathrm{cos}^{2}\\left(t\\right)+\\mathrm{sin}^{2}\\left(t\\right)&amp;=1\\\\ \\mathrm{cos}^{2}\\left(t\\right)+\\frac{9}{49}&amp;=1\\\\\\mathrm{cos}^{2}\\left(t\\right)&amp;=\\frac{40}{49}\\\\\\mathrm{cos}\\left(t\\right)&amp;=\u00b1\\sqrt[\\leftroot{1}\\uproot{2} ]{\\frac{40}{49}}=\u00b1\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{40}}{7}=\u00b1\\frac{2\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{10}}{7}\\end{align*}[\/latex][latex]\\\\[\/latex]\r\n\r\n&nbsp;\r\n\r\nBecause the angle is in the second quadrant, we know the x-value is a negative real number, so the cosine is also negative.\u00a0\u00a0So [latex] \\mathrm{cos}\\left(t\\right)=-\\frac{2\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{10}}{7}.[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137551557\" class=\"precalculus tryit\">\r\n<h3>Try It #3<\/h3>\r\n<div id=\"ti_05_02_03\">\r\n<div id=\"fs-id1165137414608\">\r\n<p id=\"fs-id1165137414609\">If [latex]\\mathrm{cos}\\left(t\\right)=\\frac{24}{25}[\/latex] and [latex]t[\/latex] is in the fourth quadrant, find [latex]\\text{sin}\\left(t\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135678598\">[reveal-answer q=\"fs-id1165135678598\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135678598\"]\r\n<p id=\"fs-id1165135678599\">[latex]\\mathrm{sin}\\left(t\\right)=-\\frac{7}{25}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137534740\" class=\"bc-section section\">\r\n<h3>Finding Sines and Cosines of Special Angles<\/h3>\r\n<p id=\"fs-id1165134340068\">We have already learned some properties of the special angles, such as the conversion from radians to degrees. In section 3.1, we also calculated sines and cosines of the special angles using the <span class=\"no-emphasis\">Pythagorean Identity<\/span> and our knowledge of triangles.<\/p>\r\n\r\n<div id=\"fs-id1165137727626\" class=\"bc-section section\">\r\n<h4>Finding Sines and Cosines of 45\u00b0 Angles and\u00a030\u00b0 and 60\u00b0 Angles<\/h4>\r\n<\/div>\r\n<div id=\"fs-id1165135435759\" class=\"bc-section section\">\r\n<p id=\"fs-id1165134394619\">We have already found the cosine and sine values for all of the most commonly encountered angles in the first quadrant of the unit circle. <a class=\"autogenerated-content\" href=\"#Table_05_02_01\">Figure 9<\/a> summarizes these values.<\/p>\r\n\r\n<table id=\"Table_05_02_01\" summary=\"..\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>Angle<\/strong><\/td>\r\n<td class=\"border\">0<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{6},[\/latex] or 30\u00b0<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{4},[\/latex] or 45\u00b0<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{3},[\/latex] or 60\u00b0<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{2},[\/latex] or 90\u00b0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>Cosine<\/strong><\/td>\r\n<td class=\"border\">1<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>Sine<\/strong><\/td>\r\n<td class=\"border\">0<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"Figure_05_02_021\" class=\"wp-caption aligncenter\" style=\"width: 673px\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"370\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132254\/CNX_Precalc_Figure_05_02_021n.jpg\" alt=\"Graph of a quarter circle with angles of 0, 30, 45, 60, and 90 degrees inscribed. Equivalence of angles in radians shown. Points along circle are marked.\" width=\"370\" height=\"330\" \/> <strong>Figure 9:\u00a0<\/strong> Graph of a quarter circle with angles of 0, 30, 45, 60, and 90 degrees inscribed. Equivalence of angles in radians shown.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137425870\" class=\"bc-section section\">\r\n<h4>Using a Calculator to Find Sine and Cosine<\/h4>\r\n<p id=\"fs-id1165137425875\">To find the cosine and sine of angles other than the <span class=\"no-emphasis\">special angles<\/span>, we turn to a computer or calculator. <strong>Be aware<\/strong>: Most calculators can be set into \u201cdegree\u201d or \u201cradian\u201d mode, which tells the calculator the units for the input value. When we evaluate [latex]\\mathrm{cos}\\left(30\\right)[\/latex] on our calculator, it will evaluate it as the cosine of 30 degrees if the calculator is in degree mode, or the cosine of 30 radians if the calculator is in radian mode.<\/p>\r\n\r\n<div id=\"fs-id1165137892278\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135415735\"><strong>Given an angle in radians, use a graphing calculator to find the cosine. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137534102\" type=\"1\">\r\n \t<li>If the calculator has degree mode and radian mode, set it to radian mode.<\/li>\r\n \t<li>Press the COS key.<\/li>\r\n \t<li>Enter the radian value of the angle and press the close-parentheses key \")\".<\/li>\r\n \t<li>Press ENTER.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_02_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137827676\">\r\n<div id=\"fs-id1165137812122\">\r\n<h3>Example 4:\u00a0 Using a Graphing Calculator to Find Sine and Cosine<\/h3>\r\n<p id=\"fs-id1165135439982\">Evaluate [latex]\\mathrm{cos}\\left(5.1\\right)[\/latex] using a graphing calculator or computer.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135436451\">[reveal-answer q=\"fs-id1165135436451\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135436451\"]\r\n<div id=\"fs-id1165131968061\">\r\n<p id=\"fs-id1165137530715\">Make sure your calculator is in radian mode.\u00a0 Enter the following keystrokes:<\/p>\r\n<p style=\"text-align: center\">\u00a0[latex]\\text{COS }\\left(5.1\\right)\\text{ ENTER}[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135190973\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\mathrm{cos}\\left(5.1\\right)\\approx0.37798[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135190240\" class=\"bc-section section\">\r\n<div class=\"textbox tryit\">\r\n<h3>Try It #4<\/h3>\r\n<div id=\"fs-id1165137416877\">\r\n<p id=\"fs-id1165137416878\">Evaluate [latex]\\mathrm{sin}\\left(2.3\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137733611\">[reveal-answer q=\"fs-id1165137733611\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137733611\"]\r\n<p id=\"fs-id1165137733612\">approximately 0.74571<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<h3>Reference Angles<\/h3>\r\n<p id=\"fs-id1165135586345\">For any given angle in the first quadrant, there is an angle in the second quadrant with the same <em>y<\/em>-value and therefore the same sine value. Because the sine value is the <em>y<\/em>-coordinate on the unit circle, the other angle with the same sine will share the same <em>y<\/em>-value, but have the opposite <em>x<\/em>-value. Therefore, its cosine value will be the opposite of the first angle\u2019s cosine value.<\/p>\r\n<p id=\"fs-id1165137469869\">Likewise, there will be an angle in the fourth quadrant with the same <em>x<\/em>-value and therefore the same cosine as the original angle in the first quadrant. The angle with the same cosine will share the same <em>x<\/em>-value but will have the opposite <em>y<\/em>-value. Therefore, its sine value will be the opposite of the original angle\u2019s sine value.<\/p>\r\n<p id=\"fs-id1165137418799\">As shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_014\">Figure 11<\/a>, angle [latex]\\alpha [\/latex] has the same sine value as angle [latex]t;[\/latex] the cosine values are opposites. Angle [latex]\\beta [\/latex] has the same cosine value as angle [latex]t;[\/latex] the sine values are opposites.<\/p>\r\n\r\n<div id=\"fs-id1165137433119\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{array}{lll}\\mathrm{sin}\\left(t\\right)=\\text{ }\\mathrm{sin}\\left(\\alpha \\right)\\text{ }\\hfill &amp; \\text{and}\\hfill &amp; \\mathrm{cos}\\left(t\\right)=-\\mathrm{cos}\\left(\\alpha \\right)\\hfill \\\\ \\mathrm{sin}\\left(t\\right)=-\\mathrm{sin}\\left(\\beta \\right)\\hfill &amp; \\text{and}\\hfill &amp; \\mathrm{cos}\\left(t\\right)=\\text{ }\\mathrm{cos}\\left(\\beta \\right)\\hfill \\end{array}[\/latex]<\/div>\r\n\r\n[caption id=\"attachment_2037\" align=\"aligncenter\" width=\"740\"]<img class=\"wp-image-2037 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/23153603\/3-3-reference-angle.png\" alt=\"\" width=\"740\" height=\"335\" \/> Figure 11[\/caption]\r\n<h4>Finding Reference Angles<\/h4>\r\n<p id=\"fs-id1165135191896\">An angle\u2019s <strong>reference angle<\/strong> is the measure of the smallest, positive, acute angle [latex]t[\/latex] formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants.<\/p>\r\n\r\n<div id=\"fs-id1165135517171\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\nAn angle\u2019s <strong>reference angle<\/strong> is the size of the smallest acute angle, [latex]{t}^{\\prime },[\/latex] formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis.\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165134258657\">We can see that reference angle is always an angle between [latex]0^{\\circ}[\/latex] and [latex]90^{\\circ},[\/latex] or [latex]0[\/latex] and [latex]\\frac{\\pi }{2}[\/latex] radians. As we can see from <a class=\"autogenerated-content\" href=\"#Figure_05_02_020\">Figure 12<\/a>, for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.<\/p>\r\n\r\n<div id=\"Figure_05_02_020\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"742\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132303\/CNX_Precalc_Figure_05_01_019-1.jpg\" alt=\"Four side by side graphs. First graph shows an angle of t in quadrant 1 in it's normal position. Second graph shows an angle of t in quadrant 2 due to a rotation of pi minus t. Third graph shows an angle of t in quadrant 3 due to a rotation of t minus pi. Fourth graph shows an angle of t in quadrant 4 due to a rotation of two pi minus t.\" width=\"742\" height=\"252\" \/> <strong>Figure 12:\u00a0<\/strong>\u00a0Reference Angles[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134122927\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165133103951\"><strong>Given an angle between [latex]0[\/latex] and [latex]2\\pi,[\/latex] or between [latex]0^{\\circ}[\/latex] and [latex]360^{\\circ},[\/latex] find its reference angle.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137696399\" type=\"1\">\r\n \t<li>An angle in the first quadrant is its own reference angle.<\/li>\r\n \t<li>For an angle in the second quadrant, the reference angle is [latex]\\pi-t [\/latex] or [latex]180^{\\circ}-t.[\/latex]<\/li>\r\n \t<li>For an angle in the third quadrant, the reference angle is\u00a0[latex]t-\\pi[\/latex] or [latex]t-180^{\\circ}.[\/latex]<\/li>\r\n \t<li>For an angle in the fourth quadrant, the reference angle is [latex]2\\pi -t[\/latex] or [latex]360^{\\circ}-t.[\/latex]<\/li>\r\n<\/ol>\r\n<strong>Given an angle less than [latex]0[\/latex] or greater than [latex]2\\pi,[\/latex] or less than [latex]0^{\\circ}[\/latex] or greater than [latex]360^{\\circ},[\/latex] find its reference angle.<\/strong>\r\n<ol id=\"fs-id1165137696399\" type=\"1\">\r\n \t<li>If an angle is less than [latex]0[\/latex] or greater than [latex]2\\pi,[\/latex] add or subtract [latex]2\\pi [\/latex] as many times as needed to find a coterminal angle between [latex]0[\/latex] and [latex]2\\pi .[\/latex]\u00a0 Similarly, if working in degrees add or subtract [latex]360^{\\circ}[\/latex] as many times as needed to find a coterminal angle between [latex]0^{\\circ}[\/latex] and [latex]360^{\\circ}.[\/latex]<\/li>\r\n \t<li>Once you have an angle\u00a0between [latex]0[\/latex] and [latex]2\\pi,[\/latex] or\u00a0between [latex]0^{\\circ}[\/latex] and [latex]360^{\\circ},[\/latex] follow steps 1 - 4 above.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_02_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137455785\">\r\n<div id=\"fs-id1165137455788\">\r\n<h3>Example 5:\u00a0 Finding a Reference Angle<\/h3>\r\n<p id=\"fs-id1165135436617\">Find the reference angle of [latex]225^{\\circ}[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_016\">Figure 13<\/a>.<\/p>\r\n\r\n<div id=\"Figure_05_02_016\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"370\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132307\/CNX_Precalc_Figure_05_02_016.jpg\" alt=\"Graph of circle with 225 degree angle inscribed.\" width=\"370\" height=\"291\" \/> <strong>Figure 13:\u00a0<\/strong>\u00a0Graph of circle with 225 degree angle inscribed.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137535343\">[reveal-answer q=\"fs-id1165137535343\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137535343\"]\r\n<p id=\"fs-id1165137535345\" style=\"text-align: left\">Because [latex]225^{\\circ}[\/latex] is in the third quadrant, the reference angle is [latex] 225^{\\circ} -180^{\\circ}=45^{\\circ}.[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137761979\" class=\"precalculus tryit\">\r\n<h3>Try It # 5<\/h3>\r\n<div id=\"ti_05_02_05\">\r\n<div id=\"fs-id1165137598484\">\r\n<p id=\"fs-id1165137598486\">Find the reference angle of [latex]\\frac{5\\pi }{3}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137806750\">[reveal-answer q=\"fs-id1165137806750\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137806750\"]\r\n<p id=\"fs-id1165137806751\">[latex]\\frac{\\pi }{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135458607\" class=\"bc-section section\">\r\n<h4 id=\"fs-id1165135477487\"><span style=\"color: #6c64ad;font-size: 1em;font-weight: 600\">Using Reference Angles to Find Exact Values for Cosine and Sine<\/span><\/h4>\r\n<div id=\"fs-id1165134190698\" class=\"bc-section section\">\r\n<p id=\"fs-id1165134259243\">We can find the cosine and sine of any angle in any quadrant if we know the cosine or sine of its reference angle. The absolute values of the cosine and sine of an angle are the same as those of the reference angle. The sign depends on the quadrant of the original angle. The cosine will be positive or negative depending on the sign of the <em>x<\/em>-values in that quadrant. The sine will be positive or negative depending on the sign of the <em>y<\/em>-values in that quadrant.<\/p>\r\nFor an angle which has a special angle as the reference angle ([latex]\\frac{\\pi}{6}, \\frac{\\pi}{4},[\/latex] or [latex]\\frac{\\pi}{3},[\/latex]) we can produce exact value outputs for sine and cosine.\r\n<div id=\"fs-id1165137401083\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165134047575\"><strong>Given an angle in standard position, find the reference angle, and the cosine and sine of the original angle. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137530558\" type=\"1\">\r\n \t<li>Measure the angle between the terminal side of the given angle and the horizontal axis. That is the reference angle.<\/li>\r\n \t<li>Determine the values of the cosine and sine of the reference angle.<\/li>\r\n \t<li>Give the cosine the same sign as the <em>x<\/em>-values in the quadrant of the original angle.<\/li>\r\n \t<li>Give the sine the same sign as the <em>y<\/em>-values in the quadrant of the original angle.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_02_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135521301\">\r\n<div id=\"fs-id1165131990665\">\r\n<h3>Example 6:\u00a0 Using Reference Angles to Find Sine and Cosine<\/h3>\r\n<ol>\r\n \t<li>Using a reference angle, find the exact values of [latex]\\mathrm{cos}\\left(150^{\\circ}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(150^{\\circ}\\right).[\/latex]<\/li>\r\n \t<li>Find angles between [latex]0^{\\circ}[\/latex] and [latex]360^{\\circ}[\/latex] which have the same exact values as [latex]\\mathrm{cos}\\left(150^{\\circ}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(150^{\\circ}\\right).[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1165135534994\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135534994\"]\r\n<ol>\r\n \t<li style=\"list-style-type: none\">\r\n<ol>\r\n \t<li style=\"text-align: left\">150\u00b0 is located in the second quadrant. The angle it makes with the x-axis is 180\u00b0 \u2212 150\u00b0 = 30\u00b0, so the reference angle is 30\u00b0.\r\nThis tells us that 150\u00b0 has the same sine and cosine values as 30\u00b0, except for the sign. We know that [latex]\\mathrm{cos}\\left(30^{\\circ}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2},[\/latex] and [latex]\\mathrm{sin}\\left(30^{\\circ}\\right)=\\frac{1}{2}.[\/latex][latex]\\\\[\/latex]Since 150\u00b0 is in the second quadrant, the x-coordinate of the point on the circle is negative, so the cosine value is negative. The y-coordinate is positive, so the sine value is positive.[latex]\\\\[\/latex]\r\n<div id=\"fs-id1165134417907\" class=\"unnumbered\" style=\"text-align: center\">\r\n<div id=\"fs-id1165134417905\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\mathrm{cos}\\left(150^{\\circ}\\right)=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<div class=\"unnumbered\" style=\"text-align: center\">[latex]\\mathrm{sin}\\left(150^{\\circ}\\right)=\\frac{1}{2}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<\/div><\/li>\r\n \t<li>From part 1, we know that\u00a0150\u00b0 is in quadrant 2 and has a reference angle of\u00a030\u00b0.\u00a0 We also discussed that the cosine value in quadrant 2 is negative.\u00a0 The other quadrant where cosine is negative is quadrant 3, since the x coordinate of any point in quadrant 3 is negative.\u00a0 Therefore, we need an angle in quadrant 3 that has a reference angle of 30\u00b0.\u00a0[latex]\\\\[\/latex]Since we know that the reference angle in quadrant 3 is [latex]t-180^{\\circ}[\/latex] where t is an angle in standard position, the angle we want can be found by solving the equation\r\n<div id=\"fs-id1165134417908\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}30^{\\circ}&amp;=t-180^{\\circ}\\\\t&amp;=180^{\\circ}+30^{\\circ}\\\\t&amp;= 210^{\\circ}.\\\\\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\nTherefore [latex]\\mathrm{cos}\\left(150^{\\circ}\\right)=\\mathrm{cos}\\left(210^{\\circ}\\right).[\/latex] [latex]\\\\[\/latex]\r\n<div>We discussed in part 1 that the sine value in quadrant 2 is positive.\u00a0 The other quadrant where sine is positive is quadrant 1, since the y coordinate is positive in quadrant 1.<\/div>\r\nTherefore,\u00a0[latex]\\mathrm{sin}\\left(150^{\\circ}\\right)=\\mathrm{sin}\\left(30^{\\circ}\\right).[\/latex][latex]\\\\[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It #6<\/h3>\r\n<ol>\r\n \t<li>Use the reference angle of [latex]315^{\\circ}[\/latex] to find the exact values for\u00a0 [latex]\\mathrm{cos}\\left(315^{\\circ}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(315^{\\circ}\\right).[\/latex]<\/li>\r\n \t<li>Find angles between [latex]0^{\\circ}[\/latex] and [latex]360^{\\circ}[\/latex] which have the same exact values as [latex]\\mathrm{cos}\\left(315\u00b0\\right)[\/latex] and [latex]\\mathrm{sin}\\left(315^{\\circ}\\right).[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1165135193309\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135193309\"]\r\n<ol id=\"fs-id1165137647526\" type=\"a\">\r\n \t<li>The reference angle is 45\u00b0 and the angle is in the fourth quadrant.\u00a0 [latex]\\mathrm{cos}\\left(315^{\\circ}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\text{ }\\mathrm{sin}\\left(315^{\\circ}\\right)=\\frac{\u2013\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/li>\r\n \t<li>[latex]\\mathrm{cos}\\left(315^{\\circ}\\right)=\\mathrm{cos}\\left(45^{\\circ}\\right)\\\\\\mathrm{sin}\\left(315^{\\circ}\\right)=\\mathrm{sin}\\left(225^{\\circ}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Example 7:\u00a0 Using Reference Angles to Find Sine and Cosine<\/h3>\r\n<ol>\r\n \t<li>Using a reference angle, find the exact values of [latex]\\mathrm{cos}\\left(\\frac{5\\pi}{4}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(\\frac{5\\pi}{4}\\right).[\/latex]<\/li>\r\n \t<li>Find angles between [latex]0\\text{ and }2\\pi[\/latex] which have the same exact values as\u00a0[latex]\\mathrm{cos}\\left(\\frac{5\\pi}{4}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(\\frac{5\\pi}{4}\\right).[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1165135534996\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135534996\"]\r\n<ol>\r\n \t<li style=\"text-align: left\">[latex]\\frac{5\\pi }{4}[\/latex] is in the third quadrant. Its reference angle is [latex]\\frac{5\\pi }{4}-\\pi =\\frac{\\pi }{4}.[\/latex]\u00a0 The cosine and sine of [latex]\\frac{\\pi }{4}[\/latex] are both [latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}.[\/latex]\u00a0 In the third quadrant, both [latex]x[\/latex] and [latex]y[\/latex] are negative, so:[latex]\\\\[\/latex]\r\n<p style=\"text-align: center\">[latex]\\mathrm{cos}\\left(\\frac{5\\pi }{4}\\right)=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\\\[\/latex][latex]\\mathrm{sin}\\left(\\frac{5\\pi }{4}\\right)=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/p>\r\n<\/li>\r\n \t<li>\u00a0We know from part 3 that\u00a0[latex]\\frac{5\\pi }{4}[\/latex] is in the third quadrant and its reference angle is [latex]\\frac{5\\pi }{4}-\\pi =\\frac{\\pi }{4}.[\/latex]\u00a0 We also know that both cosine and sine are negative in quadrant 3.\u00a0 Cosine is also negative in quadrant 2 since the [latex]x[\/latex]-coordinate is negative in quadrant 2, so we can use the fact that in quadrant 2, the reference angle is:\r\n<div id=\"fs-id1165134417904\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\\\[\/latex][latex]\\begin{align*}\\frac{\\pi }{4}&amp;=\\pi-t &amp;&amp; \\text{t is an angle in standard position. } \\\\t&amp;=\\pi-\\frac{\\pi }{4}\\\\t&amp;=\\frac{3\\pi}{4}\\end{align*}[\/latex]\u00a0[latex]\\\\[\/latex]<\/div>\r\nWe therefore know that [latex]\\mathrm{cos}\\left(\\frac{5\\pi }{4}\\right)=\\mathrm{cos}\\left(\\frac{3\\pi }{4}\\right).[\/latex][latex]\\\\[\/latex] Sine is also negative in quadrant 4 since the [latex]y[\/latex]-coordinate is negative in quadrant 4.\u00a0 We know that in quadrant 4 the reference angle is:\r\n<div id=\"fs-id1165134417906\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}\\frac{\\pi}{4}&amp;=2\\pi-t&amp;&amp;\\text{Now solve for t.}\\\\t&amp;=2\\pi-\\frac{\\pi}{4}\\\\t&amp;=\\frac{7\\pi}{4}\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\nWe therefore know that [latex]\\mathrm{sin}\\left(\\frac{5\\pi}{4}\\right)=\\mathrm{sin}\\left(\\frac{7\\pi}{4}\\right).[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n<div id=\"fs-id1165137550671\" class=\"precalculus tryit\">\r\n<h3>Try It # 7<\/h3>\r\n<div id=\"ti_05_02_06\">\r\n<div id=\"fs-id1165134081332\">\r\n<ol id=\"fs-id1165134081334\" type=\"a\">\r\n \t<li>Use the reference angle of [latex]-\\frac{\\pi }{6}[\/latex] to find the exact values of\u00a0 [latex]\\mathrm{cos}\\left(-\\frac{\\pi}{6}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(-\\frac{\\pi}{6}\\right).[\/latex]<\/li>\r\n \t<li>Find angles between [latex]0\\text{ and }2\\pi[\/latex] which have the same exact values as\u00a0[latex]\\mathrm{cos}\\left(-\\frac{\\pi}{6}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(-\\frac{\\pi}{6}\\right).[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1165135193308\">[reveal-answer q=\"fs-id1165135193308\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135193308\"]\r\n<ol id=\"fs-id1165137647526\" type=\"a\">\r\n \t<li>The reference angle is [latex]\\frac{\\pi}{6}[\/latex] and the angle is in the fourth quadrant. [latex]\\mathrm{cos}\\left(-\\frac{\\pi }{6}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2},\\text{ }\\mathrm{sin}\\left(-\\frac{\\pi }{6}\\right)=-\\frac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]\\mathrm{cos}\\left(-\\frac{\\pi }{6}\\right)=\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)=\\mathrm{cos}\\left(\\frac{11\\pi }{6}\\right).[\/latex][latex]\\\\[\/latex][latex]\\mathrm{sin}\\left(-\\frac{\\pi}{6}\\right)=\\mathrm{sin}\\left(\\frac{7\\pi}{6}\\right)=\\mathrm{cos}\\left(\\frac{11\\pi }{6}\\right).[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137766908\" class=\"bc-section section\">\r\n<h4>Using Reference Angles to Find Coordinates<\/h4>\r\n<p id=\"fs-id1165137766914\">Now that we have learned how to find the cosine and sine values for angles whose reference angles are acute special angles, the rest of the special angles on the unit circle can be determine. They are shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_017\">Figure 14<\/a>. Take time to learn the [latex]\\left(x,y\\right)[\/latex] coordinates of all of the major angles in the first quadrant.<\/p>\r\n\r\n<div id=\"Figure_05_02_017\" class=\"wp-caption aligncenter\">\r\n<div class=\"wp-caption-text\"><\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"739\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132311\/CNX_Precalc_Figure_05_02_017.jpg\" alt=\"Graph of unit circle with angles in degrees, angles in radians, and points along the circle inscribed.\" width=\"739\" height=\"478\" \/> <strong>Figure 14:<\/strong> Special angles and coordinates of corresponding points on the unit circle[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165135704914\">In addition to learning the values for special angles, we can use reference angles to find [latex]\\left(x,y\\right)[\/latex] coordinates of any point on the unit circle, using what we know of reference angles along with the <span class=\"no-emphasis\">identities<\/span><\/p>\r\n\r\n<div id=\"eip-150\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}x&amp;=\\mathrm{cos}\\left(t\\right) \\\\ y&amp;=\\mathrm{sin}\\left(t\\right).\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137411099\">First we find the reference angle corresponding to the given angle. Then we take the sine and cosine values of the <span class=\"no-emphasis\">reference angle<\/span>, and give them the signs corresponding to the <em>y<\/em>- and <em>x<\/em>-values of the quadrant.<\/p>\r\n\r\n<div id=\"fs-id1165137676498\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135381357\"><strong>Given the angle of a point on a unit circle, find the [latex]\\left(x,y\\right)[\/latex] coordinates of the point.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165134211369\" type=\"1\">\r\n \t<li>Find the reference angle by measuring the smallest angle to the <em>x<\/em>-axis.<\/li>\r\n \t<li>Find the cosine and sine of the reference angle.<\/li>\r\n \t<li>Determine the appropriate signs for [latex]x[\/latex] and [latex]y[\/latex] in the given quadrant.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_02_07\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137409939\">\r\n<div id=\"fs-id1165137409941\">\r\n<h3>Example 8:\u00a0 Using the Unit Circle to Find Coordinates<\/h3>\r\n<p id=\"fs-id1165137409947\">Find the coordinates of the point on the unit circle at an angle of [latex]\\frac{7\\pi }{6}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137704533\">[reveal-answer q=\"fs-id1165137704533\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137704533\"]\r\n<p id=\"fs-id1165137704536\">We know that the angle [latex]\\frac{7\\pi }{6}[\/latex] is in the third quadrant.<\/p>\r\n<p id=\"fs-id1165137851856\">First, let\u2019s find the reference angle by measuring the angle to the <em>x<\/em>-axis. To find the reference angle of an angle whose terminal side is in quadrant III, we find the difference of the angle and [latex]\\pi .[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165134555590\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\frac{7\\pi }{6}-\\pi =\\frac{\\pi }{6}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137394109\">Next, we will find the cosine and sine of the reference angle:<\/p>\r\n\r\n<div id=\"fs-id1165137394112\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)=\\frac{1}{2}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165132011271\">We must determine the appropriate signs for <em>x<\/em> and <em>y<\/em> in the given quadrant. Because our original angle is in the third quadrant, where both [latex]x[\/latex] and [latex]y[\/latex] are negative, both cosine and sine are negative.<\/p>\r\n\r\n<div id=\"fs-id1165137758918\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}\\mathrm{cos}\\left(\\frac{7\\pi }{6}\\right)&amp;=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2} \\\\ \\mathrm{sin}\\left(\\frac{7\\pi }{6}\\right)&amp;=-\\frac{1}{2}\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135503862\">Now we can calculate the [latex]\\left(x,y\\right)[\/latex] coordinates using the identities [latex]x=\\mathrm{cos}\\left(\\theta\\right) [\/latex] and [latex]y=\\mathrm{sin}\\left(\\theta\\right) .[\/latex]<\/p>\r\n<p id=\"fs-id1165137411025\">The coordinates of the point are [latex]\\left(-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2},-\\frac{1}{2}\\right)[\/latex] on the unit circle.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137407555\" class=\"precalculus tryit\">\r\n<h3>Try It # 8<\/h3>\r\n<div id=\"ti_05_02_07\">\r\n<div id=\"fs-id1165137535595\">\r\n<p id=\"fs-id1165137535596\">Find the coordinates of the point on the unit circle at an angle of [latex]\\frac{5\\pi }{3}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135530358\">[reveal-answer q=\"fs-id1165135530358\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135530358\"]\r\n<p id=\"fs-id1165135530359\">[latex]\\left(\\frac{1}{2},-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137583095\" class=\"precalculus media\">\r\n<h3>Circles with Radius Different Than 1<\/h3>\r\nSuppose we have a circle centered at the origin other than the unit circle. How can we find the coordinates of a point where the terminal side of the angle in standard position intersects the circle? We know from Section 3.1 that the sine and cosine values of an acute angle do not change regardless of the size of the triangle. This means that [latex]\\mathrm{cos}\\left(t\\right)[\/latex] will be the same whether [latex]t[\/latex] is in a triangle with small or large sides. However, it is clear that the coordinates of the point of intersection of the terminal side with the circle of radius r will not be the same as the coordinates of the point on the unit circle.\r\n\r\nWe can apply similar reasoning to considering values of sine and cosine values on a circle that is not a unit circle as we did earlier in this section. By drawing an angle in standard position in quadrant 1, we can drop a perpendicular from the point of intersection of the terminal side with the circle whose center is at the origin to the x axis. If the point on the circle is (x, y), then the sides of the right triangle formed by dropping the perpendicular will also be [latex]x\\text{ and }y.[\/latex] Now, instead of the radius being 1, we would have a radius of [latex]r.[\/latex] Therefore, [latex]\\mathrm{cos}\\left(t\\right)=\\frac{x}{r}\\text{ and }\\mathrm{sin}\\left(t\\right)=\\frac{y}{r}.[\/latex] This means that [latex]x=r\\mathrm{cos}\\left(t\\right)\\text{ and }y=r\\mathrm{sin}\\left(t\\right).[\/latex]\r\n<div class=\"textbox Definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165137793724\">A point [latex]\\left(x,y\\right)[\/latex] on the circle, centered at the origin, with radius [latex]r[\/latex] corresponds to<\/p>\r\n\r\n<div id=\"Equation_05_02_01\" style=\"text-align: center\">[latex]\\begin{align*}x&amp;=r\\mathrm{cos}\\left(t\\right)\\text{ and }\\\\y&amp;=r\\mathrm{sin}\\left(t\\right)\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div>where [latex]t[\/latex] is the angle in standard position.<\/div>\r\n<\/div>\r\n<div id=\"Equation_05_02_02\">\r\n<div id=\"fs-id1165137704533\">\r\n<div class=\"textbox examples\">\r\n<h3>Example 9: \u00a0Finding Coordinates on a Circle with Radius [latex]r[\/latex]<\/h3>\r\nFind the coordinates of the point on the circle with radius 5 at an angle of [latex]\\frac{7\\pi}{6}.[\/latex]\r\n\r\n[reveal-answer q=\"fs-id1165137409939\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137409939\"]\r\nWe already know from Example 7 that:\r\n<p style=\"text-align: center\">[latex]\\begin{align*}\\mathrm{cos}\\left(\\frac{7\\pi }{6}\\right)&amp;=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2} \\\\ \\mathrm{sin}\\left(\\frac{7\\pi }{6}\\right)&amp;=-\\frac{1}{2}\\end{align*}[\/latex][latex]\\\\[\/latex]<\/p>\r\n<p id=\"fs-id1165135503862\" style=\"text-align: left\">Now we can calculate the [latex]\\left(x,y\\right)[\/latex] coordinates using the identities<\/p>\r\n<p style=\"text-align: center\">[latex]x=r\\mathrm{cos}\\left(\\theta\\right) [\/latex] and [latex]y=r\\mathrm{sin}\\left(\\theta\\right) .[\/latex]<\/p>\r\n<p id=\"fs-id1165137411025\">The coordinates of the point are [latex]\\left(-\\frac{5\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2},-\\frac{5}{2}\\right)[\/latex] on the circle with radius 5.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137407555\" class=\"precalculus tryit\">\r\n<h3>Try It # 9<\/h3>\r\n<div id=\"ti_05_02_07\">\r\n<div id=\"fs-id1165137535595\">\r\n<p id=\"fs-id1165137535596\">Find the coordinates of the point on the circle with radius 7 at an angle of [latex]\\frac{5\\pi }{3}.[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165135530360\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135530360\"]\r\n<p id=\"fs-id1165135530359\">[latex]\\left(\\frac{7}{2},-\\frac{7\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h3>Even and Odd Functions<\/h3>\r\n<p id=\"fs-id1165137571788\">To be able to use our sine and cosine functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.<\/p>\r\nRecall that:\r\n<ul>\r\n \t<li id=\"fs-id1165134042164\">An <span class=\"no-emphasis\">even function<\/span> is one in which [latex]f\\left(-x\\right)=f\\left(x\\right).[\/latex]<\/li>\r\n \t<li id=\"fs-id1165137443964\">An <span class=\"no-emphasis\">odd function<\/span> is one in which [latex]f\\left(-x\\right)=-f\\left(x\\right).[\/latex]<\/li>\r\n<\/ul>\r\nWe can test whether a trigonometric function is even or odd by drawing a <span class=\"no-emphasis\">unit circle<\/span> with a positive and a negative angle, as in <a class=\"autogenerated-content\" href=\"#Figure_05_03_007\">\u00a0Figure 15.\u00a0<\/a>The sine of the positive angle is [latex]y.[\/latex] The sine of the negative angle is [latex]-y.[\/latex]The <span class=\"no-emphasis\">sine function<\/span>, then, is an odd function. \u00a0The cosine of the positive angle is [latex]x[\/latex], as is the cosine of the negative angle. \u00a0Therefore, the cosine function is an even function.\r\n<div id=\"Figure_05_03_007\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"350\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132429\/CNX_Precalc_Figure_05_03_007.jpg\" alt=\"Graph of circle with angle of t and -t inscribed. Point of (x, y) is at intersection of terminal side of angle t and edge of circle. Point of (x, -y) is at intersection of terminal side of angle -t and edge of circle.\" width=\"350\" height=\"265\" \/> Figure 15[\/caption]\r\n\r\n<\/div>\r\nWe can summarize this by saying:\r\n<p style=\"text-align: center\">[latex]\\begin{align*}\\mathrm{sin}\\left(-x\\right)&amp;=-\\mathrm{sin}\\left(x\\right)\\\\ \\mathrm{cos}\\left(-x\\right)&amp;=\\mathrm{cos}\\left(x\\right)\\end{align*}[\/latex]<\/p>\r\n<p id=\"fs-id1165137828452\">Access these online resources for additional instruction and practice with sine and cosine functions.<\/p>\r\n\r\n<ul id=\"fs-id1165135160446\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/trigunitcir\">Trigonometric Functions Using the Unit Circle<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/sincosuc\">Sine and Cosine from the Unit Circle<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/sincosmult\">Sine and Cosine from the Unit Circle and Multiples of Pi Divided by Six<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/sincosmult4\">Sine and Cosine from the Unit Circle and Multiples of Pi Divided by Four<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/trigrefang\">Trigonometric Functions Using Reference Angles<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165133238450\" class=\"key-equations\">\r\n<h3>Key Equations<\/h3>\r\n<table id=\"eip-id1165134284283\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">Cosine<\/td>\r\n<td class=\"border\">[latex]\\mathrm{cos}\\left(t\\right)=x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Sine<\/td>\r\n<td class=\"border\">[latex]\\mathrm{sin}\\left(t\\right)=y[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Pythagorean Identity<\/td>\r\n<td class=\"border\">[latex]{\\mathrm{cos}}^{2}\\left(t\\right)+{\\mathrm{sin}}^{2}\\left(t\\right)=1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137692629\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1165137772132\">\r\n \t<li>Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of 1 unit.<\/li>\r\n \t<li>Using the unit circle, the sine of an angle [latex]t[\/latex] equals the <em>y<\/em>-value of the endpoint on the unit circle of an arc of length [latex]t[\/latex] whereas the cosine of an angle [latex]t[\/latex] equals the <em>x<\/em>-value of the endpoint.<\/li>\r\n \t<li>When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles.<\/li>\r\n \t<li>Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known.<\/li>\r\n \t<li>The domain of the sine and cosine functions is all real numbers.<\/li>\r\n \t<li>The range of both the sine and cosine functions is [latex]\\left[-1,1\\right].[\/latex]<\/li>\r\n \t<li>The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.<\/li>\r\n \t<li>The signs of the sine and cosine are determined from the <em>x<\/em>- and <em>y<\/em>-values in the quadrant of the original angle.<\/li>\r\n \t<li>An angle\u2019s reference angle is the size angle, [latex]t,[\/latex] formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis.<\/li>\r\n \t<li>Reference angles can be used to find the sine and cosine of the original angle.<\/li>\r\n \t<li>Reference angles can also be used to find the coordinates of a point on a unit circle.<\/li>\r\n \t<li>When the radius of a circle centered at the origin is not 1, we can find coordinates of a point on the circle by multiplying the sine and cosine of the angle by [latex]r.[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl id=\"fs-id1165137762325\">\r\n \t<dt>cosine function<\/dt>\r\n \t<dd id=\"fs-id1165137408620\">the <em>x<\/em>-value of the point on a unit circle corresponding to a given angle<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137812230\">\r\n \t<dt>Pythagorean Identity<\/dt>\r\n \t<dd id=\"fs-id1165137812235\">a corollary of the Pythagorean Theorem stating that the square of the cosine of a given angle plus the square of the sine of that angle equals 1<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137812240\">\r\n \t<dt>sine function<\/dt>\r\n \t<dd id=\"fs-id1165137749964\">the <em>y<\/em>-value of the point on a unit circle corresponding to a given angle<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137749972\">\r\n \t<dt>unit circle<\/dt>\r\n \t<dd id=\"fs-id1165135693774\">a circle with a center at [latex]\\left(0,0\\right)[\/latex] and radius 1.<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Find function values for the sine and cosine of [latex]30^{\\circ}\\text{ or }\\left(\\frac{\\pi }{6}\\right),\\text{ }45^{\\circ}\\text{ or }\\left(\\frac{\\pi }{4}\\right)[\/latex] and [latex]{60^{\\circ}}\\text{or}\\left(\\frac{\\pi }{3}\\right).[\/latex]<\/li>\n<li>Identify the domain and range of sine and cosine functions.<\/li>\n<li>Use reference angles to evaluate trigonometric functions.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Figure_05_02_001\" class=\"small\">\n<div style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132215\/CNX_Precalc_Figure_05_02_001.jpg\" alt=\"Photo of a ferris wheel.\" width=\"370\" height=\"246\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1:<\/strong> The Singapore Flyer is the world\u2019s tallest Ferris wheel. (credit: \u201cVibin JK\u201d\/Flickr)<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137692173\">Looking for a thrill? Then consider a ride on the Singapore Flyer, the world\u2019s tallest Ferris wheel. Located in Singapore, the Ferris wheel soars to a height of 541 feet\u2014a little more than a tenth of a mile! Described as an observation wheel, riders enjoy spectacular views as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs.<\/p>\n<div id=\"fs-id1165137714296\" class=\"bc-section section\">\n<h3>Finding Function Values for the Sine and Cosine<\/h3>\n<p id=\"fs-id1165134153045\">To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1, as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_002\">Figure 2<\/a>. The angle (in radians) that [latex]t[\/latex] intercepts forms an arc of length [latex]s.[\/latex] Using the formula [latex]s=rt,[\/latex] and knowing that [latex]r=1,[\/latex] we see that for a <span class=\"no-emphasis\">unit circle<\/span>, [latex]s=t.[\/latex]<\/p>\n<p id=\"fs-id1165137428004\">Recall that the <em>x- <\/em>and <em>y-<\/em>axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV.<\/p>\n<p id=\"fs-id1165137837083\">For any angle [latex]t,[\/latex] we can label the intersection of the terminal side and the unit circle by its coordinates, [latex]\\left(x,y\\right).[\/latex] \u00a0 Consider an angle that is in the first quadrant.\u00a0 We can drop a perpendicular line to the x-axis to create a right triangle. The sides of the right triangle will be x and y. \u00a0 If we use our right trigonometric definitions from Section 3.1, we can see that [latex]\\mathrm{cos}\\left(t\\right)=\\frac{x}{1}[\/latex] and [latex]\\mathrm{sin}\\left(t\\right)=\\frac{y}{1}.[\/latex]\u00a0 This means [latex]\\left(x,\\text{ }y\\right)\\text{ }=\\text{ }\\left(\\mathrm{cos}\\left(t\\right),\\text{ }\\mathrm{sin}\\left(t\\right)\\right).[\/latex]<\/p>\n<div id=\"Figure_05_02_002\" class=\"small\">\n<div style=\"width: 381px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132218\/CNX_Precalc_Figure_05_02_002.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and an arc created by the angle with length s. The terminal side of the angle intersects the circle at the point (x,y).\" width=\"371\" height=\"293\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 2:<\/strong> Unit circle where the central angle is [latex]t[\/latex] radians<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137638636\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165135560820\">A <strong>unit circle<\/strong> has a center at [latex]\\left(0,0\\right)[\/latex] and radius [latex]1[\/latex] . In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle [latex]t\\text{.}[\/latex]<\/p>\n<p id=\"fs-id1165137714560\">Let [latex]\\left(x,y\\right)[\/latex] be the endpoint on the unit circle of an arc of arc length [latex]t.[\/latex] The [latex]\\left(x,y\\right)[\/latex] coordinates of this point can be described as functions of the angle [latex]t[\/latex] where:<\/p>\n<p style=\"text-align: center\">\u00a0 [latex]\\mathrm{cos}\\left(t\\right)=x[\/latex] and<\/p>\n<p style=\"text-align: center\">[latex]\\mathrm{sin}\\left(t\\right)=y.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137726457\" class=\"bc-section section\">\n<p>Note that this definition allows us to use angles which are not acute or in other words, angles in standard position whose terminal side is not in the first quadrant.<\/p>\n<p id=\"fs-id1165137768253\">Now that we have our unit circle labeled, we can learn how the [latex]\\left(x,y\\right)[\/latex] coordinates relate to the <span class=\"no-emphasis\">arc length <\/span>and <span class=\"no-emphasis\">angle<\/span>. The sine function relates a real number [latex]t[\/latex] to the <em>y<\/em>-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle [latex]t[\/latex] equals the <em>y<\/em>-value of the endpoint on the unit circle of an arc of length [latex]t.[\/latex] In <a class=\"autogenerated-content\" href=\"#Figure_05_02_002\">Figure 3<\/a>, the sine is equal to [latex]y.[\/latex] Like all functions, the sine function has an input and an output. Its input is the measure of the angle; its output is the <em>y<\/em>-coordinate of the corresponding point on the unit circle.<\/p>\n<p id=\"fs-id1165137748691\">The cosine function of an angle [latex]t[\/latex] equals the <em>x<\/em>-value of the endpoint on the unit circle of an arc of length [latex]t.[\/latex] In <a class=\"autogenerated-content\" href=\"#Figure_05_02_003\">Figure 3<\/a>, the cosine is equal to [latex]x.[\/latex]<\/p>\n<div id=\"Figure_05_02_003\" class=\"small\">\n<div style=\"width: 381px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132221\/CNX_Precalc_Figure_05_02_003.jpg\" alt=\"Illustration of an angle t, with terminal side length equal to 1, and an arc created by angle with length t. The terminal side of the angle intersects the circle at the point (x,y), which is equivalent to (cos t, sin t).\" width=\"371\" height=\"172\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 3:\u00a0<\/strong> Illustration of an angle t, with terminal side length equal to 1, and an arc created by angle with length t.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137463311\"><strong>Important Note:\u00a0\u00a0<\/strong>Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses: [latex]\\mathrm{sin}\\text{ }t[\/latex] is the same as [latex]\\mathrm{sin}\\left(t\\right)[\/latex] and [latex]\\mathrm{cos}\\text{ }t[\/latex] is the same as [latex]\\mathrm{cos}\\left(t\\right).[\/latex] Likewise, [latex]{\\mathrm{cos}}^{2}\\left(t\\right)[\/latex] is a commonly used shorthand notation for [latex]{\\left(\\mathrm{cos}\\left(t\\right)\\right)}^{2}.[\/latex] Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer.<\/p>\n<div id=\"fs-id1165135251444\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165131968080\"><strong>Given a point <em>P\u00a0<\/em>[latex]\\left(x,y\\right)[\/latex] on the unit circle corresponding to an angle of [latex]t,[\/latex] find the sine and cosine.<\/strong><\/p>\n<ol id=\"fs-id1165134108592\" type=\"1\">\n<li>The sine of [latex]t[\/latex] is equal to the <em>y<\/em>-coordinate of point [latex]P:\\mathrm{sin}\\left(t\\right)=y.[\/latex]<\/li>\n<li>The cosine of [latex]t[\/latex] is equal to the <em>x<\/em>-coordinate of point [latex]P: \\text{cos}\\left(t\\right)=x.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_02_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137417674\">\n<div id=\"fs-id1165137824250\">\n<h3>Example 1:\u00a0 Finding Function Values for Sine and Cosine<\/h3>\n<p id=\"fs-id1165137463869\">Point [latex]P[\/latex] is a point on the unit circle corresponding to an angle of [latex]t,[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_004\">Figure 4<\/a>. Find [latex]\\mathrm{cos}\\left(t\\right)[\/latex] and [latex]\\mathrm{sin}\\left(t\\right).[\/latex]<\/p>\n<div id=\"Figure_05_02_004\" class=\"small\">\n<div style=\"width: 381px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132224\/CNX_Precalc_Figure_05_02_004.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (1\/2, square root of 3 over 2).\" width=\"371\" height=\"293\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 4:\u00a0<\/strong> Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the given point<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137436467\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137436467\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137436467\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135417749\">We know that [latex]\\mathrm{cos}\\left(t\\right)[\/latex] is the <em>x<\/em>-coordinate of the corresponding point on the unit circle and [latex]\\mathrm{sin}\\left(t\\right)[\/latex] is the <em>y<\/em>-coordinate of the corresponding point on the unit circle. So:<\/p>\n<div id=\"eip-id1165135678435\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}x&=\\mathrm{cos}\\left(t\\right)=\\frac{1}{2}\\\\ y&=\\mathrm{sin}\\left(t\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2} \\end{align*}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137723202\" class=\"precalculus tryit\">\n<h3>Try It #1<\/h3>\n<div id=\"ti_05_02_01\">\n<div id=\"fs-id1165137767886\">\n<p id=\"fs-id1165137767887\">A certain angle [latex]t[\/latex] corresponds to a point on the unit circle at [latex]\\left(-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}\\right)[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_005\">Figure 5<\/a>. Find [latex]\\mathrm{cos}\\left(t\\right)[\/latex] and [latex]\\mathrm{sin}\\left(t\\right).[\/latex]<\/p>\n<div id=\"Figure_05_02_005\" class=\"small\">\n<div style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132227\/CNX_Precalc_Figure_05_02_005.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (negative square root of 2 over 2, square root of 2 over 2).\" width=\"370\" height=\"291\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 5:\u00a0<\/strong> Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the given point.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137604785\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137604785\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137604785\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137438734\" style=\"text-align: center\">[latex]\\mathrm{cos}\\left(t\\right)=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\text{ }\\mathrm{sin}\\left(t\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135596524\" class=\"bc-section section\">\n<h4>Finding Sines and Cosines of Angles on an Axis<\/h4>\n<p id=\"fs-id1165135438852\">For <strong>quadrantral angles<\/strong>, the corresponding point on the unit circle falls on the <em>x- <\/em>or <em>y<\/em>-axis. In that case, we can easily calculate cosine and sine from the values of [latex]x[\/latex] and [latex]y.[\/latex]<\/p>\n<div id=\"Example_05_02_02\" class=\"textbox examples\">\n<div id=\"fs-id1165135173908\">\n<div id=\"fs-id1165137584111\">\n<h3>Example 2:\u00a0 Calculating Sines and Cosines Along an Axis<\/h3>\n<p id=\"fs-id1165137548354\">Find [latex]\\mathrm{cos}\\left(90^{\\circ}\\right)[\/latex] and [latex]\\text{sin}\\left(90^{\\circ}\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137418560\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137418560\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137418560\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137460268\">Moving [latex]90^{\\circ}[\/latex] counterclockwise around the unit circle from the positive <em>x<\/em>-axis brings us to the top of the circle, where the [latex]\\left(x,y\\right)[\/latex] coordinates are (0, 1), as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_006\">Figure 6<\/a>.<\/p>\n<div id=\"Figure_05_02_006\" class=\"small\">\n<div style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132230\/CNX_Precalc_Figure_05_02_006.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (0,1).\" width=\"370\" height=\"291\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 6:\u00a0<\/strong>\u00a0Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (0, 1).<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137399724\">Using our definitions of cosine and sine,<\/p>\n<div id=\"eip-id1165135209632\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}x&=\\mathrm{cos}\\left(t\\right)=\\mathrm{cos}\\left(90^{\\circ}\\right)=0\\\\ y&=\\mathrm{sin}\\left(t\\right)=\\mathrm{sin}\\left(90^{\\circ}\\right)=1\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"eip-id1165134388912\" style=\"text-align: center\">The cosine of 90\u00b0 is 0; the sine of 90\u00b0 is 1.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137405943\" class=\"precalculus tryit\">\n<h3>Try It #2<\/h3>\n<div id=\"ti_05_02_02\">\n<div id=\"fs-id1165134302493\">\n<p id=\"fs-id1165137386643\">Find cosine and sine of the angle [latex]\\pi .[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137675196\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137675196\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137675196\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134205964\">[latex]\\mathrm{cos}\\left(\\pi \\right)=-1,[\/latex]\u00a0 \u00a0[latex]\\mathrm{sin}\\left(\\pi \\right)=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137864121\" class=\"bc-section section\">\n<h4>The Pythagorean Identity<\/h4>\n<p id=\"fs-id1165134198719\">Now that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is [latex]{x}^{2}+{y}^{2}=1.[\/latex] Because [latex]x=\\mathrm{cos}\\left(t\\right)[\/latex] and [latex]y=\\mathrm{sin}\\left(t\\right),[\/latex] we can substitute for [latex]x[\/latex] and [latex]y[\/latex] to get [latex]\\mathrm{cos}^{2}\\left(t\\right)+\\mathrm{sin}^{2}\\left(t\\right)=1.[\/latex]\u00a0 This equation, [latex]\\mathrm{cos}^{2}\\left(t\\right)+\\mathrm{sin}^{2}\\left(t\\right)=1,[\/latex] is known as the <strong style=\"font-size: 1rem;text-align: initial\">Pythagorean Identity<\/strong><span style=\"font-size: 1rem;text-align: initial\">. See <\/span><a class=\"autogenerated-content\" style=\"font-size: 1rem;text-align: initial\" href=\"#Figure_05_02_007\">Figure 7<\/a><span style=\"font-size: 1rem;text-align: initial\">.<\/span><\/p>\n<div id=\"Figure_05_02_007\" class=\"small\">\n<div style=\"width: 381px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132232\/CNX_Precalc_Figure_05_02_007.jpg\" alt=\"Graph of an angle t, with a point (x,y) on the unit circle. And equation showing the equivalence of 1, x^2 + y^2, and cos^2 t + sin^2 t.\" width=\"371\" height=\"160\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 7<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137925355\">We can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or vice versa. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution.<\/p>\n<div id=\"fs-id1165137925357\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165137453137\">The <strong>Pythagorean Identity<\/strong> states that, for any real number [latex]t,[\/latex]<\/p>\n<div id=\"Equation_05_02_03\" style=\"text-align: center\">[latex]{\\mathrm{cos}}^{2}\\left(t\\right)+{\\mathrm{sin}}^{2}\\left(t\\right)=1.[\/latex]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731366\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137934414\"><strong>Given the sine of some angle [latex]t[\/latex] and its quadrant location, find the cosine of [latex]t.[\/latex] <\/strong><\/p>\n<ol id=\"fs-id1165137901198\" type=\"1\">\n<li>Substitute the known value of [latex]\\mathrm{sin}\\left(t\\right)[\/latex] into the Pythagorean Identity.<\/li>\n<li>Solve for [latex]\\mathrm{cos}\\left(t\\right).[\/latex]<\/li>\n<li>Choose the solution with the appropriate sign for the <em>x<\/em>-values in the quadrant where [latex]t[\/latex] is located.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_02_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137724522\">\n<div id=\"fs-id1165135251259\">\n<h3>Example 3:\u00a0 Finding a Cosine from a Sine or a Sine from a Cosine<\/h3>\n<p id=\"fs-id1165137434023\">If [latex]\\mathrm{sin}\\left(t\\right)=\\frac{3}{7}[\/latex] and [latex]t[\/latex] is in the second quadrant, find [latex]\\mathrm{cos}\\left(t\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137812666\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137812666\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137812666\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137405175\">If we drop a vertical line from the point on the unit circle corresponding to [latex]t,[\/latex] we create a right triangle, from which we can see that the Pythagorean Identity is simply one case of the Pythagorean Theorem. See <a class=\"autogenerated-content\" href=\"#Figure_05_02_008\">Figure 8<\/a>.<\/p>\n<div id=\"Figure_05_02_008\" class=\"small\">\n<div style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132235\/CNX_Precalc_Figure_05_02_008.jpg\" alt=\"Graph of a unit circle with an angle that intersects the circle at a point with the y-coordinate equal to 3\/7.\" width=\"370\" height=\"291\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 8:\u00a0<\/strong> Graph of a unit circle with an angle that intersects the circle at a point with the y-coordinate equal to 3\/7.<\/p>\n<\/div>\n<\/div>\n<p id=\"eip-id1165137727966\">Substituting the known value for sine into the Pythagorean Identity,<\/p>\n<div id=\"fs-id1165134393100\" class=\"unnumbered\">\n<p>[latex]\\begin{align*}\\mathrm{cos}^{2}\\left(t\\right)+\\mathrm{sin}^{2}\\left(t\\right)&=1\\\\ \\mathrm{cos}^{2}\\left(t\\right)+\\frac{9}{49}&=1\\\\\\mathrm{cos}^{2}\\left(t\\right)&=\\frac{40}{49}\\\\\\mathrm{cos}\\left(t\\right)&=\u00b1\\sqrt[\\leftroot{1}\\uproot{2} ]{\\frac{40}{49}}=\u00b1\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{40}}{7}=\u00b1\\frac{2\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{10}}{7}\\end{align*}[\/latex][latex]\\\\[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Because the angle is in the second quadrant, we know the x-value is a negative real number, so the cosine is also negative.\u00a0\u00a0So [latex]\\mathrm{cos}\\left(t\\right)=-\\frac{2\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{10}}{7}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137551557\" class=\"precalculus tryit\">\n<h3>Try It #3<\/h3>\n<div id=\"ti_05_02_03\">\n<div id=\"fs-id1165137414608\">\n<p id=\"fs-id1165137414609\">If [latex]\\mathrm{cos}\\left(t\\right)=\\frac{24}{25}[\/latex] and [latex]t[\/latex] is in the fourth quadrant, find [latex]\\text{sin}\\left(t\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135678598\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135678598\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135678598\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135678599\">[latex]\\mathrm{sin}\\left(t\\right)=-\\frac{7}{25}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137534740\" class=\"bc-section section\">\n<h3>Finding Sines and Cosines of Special Angles<\/h3>\n<p id=\"fs-id1165134340068\">We have already learned some properties of the special angles, such as the conversion from radians to degrees. In section 3.1, we also calculated sines and cosines of the special angles using the <span class=\"no-emphasis\">Pythagorean Identity<\/span> and our knowledge of triangles.<\/p>\n<div id=\"fs-id1165137727626\" class=\"bc-section section\">\n<h4>Finding Sines and Cosines of 45\u00b0 Angles and\u00a030\u00b0 and 60\u00b0 Angles<\/h4>\n<\/div>\n<div id=\"fs-id1165135435759\" class=\"bc-section section\">\n<p id=\"fs-id1165134394619\">We have already found the cosine and sine values for all of the most commonly encountered angles in the first quadrant of the unit circle. <a class=\"autogenerated-content\" href=\"#Table_05_02_01\">Figure 9<\/a> summarizes these values.<\/p>\n<table id=\"Table_05_02_01\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\"><strong>Angle<\/strong><\/td>\n<td class=\"border\">0<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{6},[\/latex] or 30\u00b0<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{4},[\/latex] or 45\u00b0<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{3},[\/latex] or 60\u00b0<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{2},[\/latex] or 90\u00b0<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>Cosine<\/strong><\/td>\n<td class=\"border\">1<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">0<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>Sine<\/strong><\/td>\n<td class=\"border\">0<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"Figure_05_02_021\" class=\"wp-caption aligncenter\" style=\"width: 673px\">\n<div style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132254\/CNX_Precalc_Figure_05_02_021n.jpg\" alt=\"Graph of a quarter circle with angles of 0, 30, 45, 60, and 90 degrees inscribed. Equivalence of angles in radians shown. Points along circle are marked.\" width=\"370\" height=\"330\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 9:\u00a0<\/strong> Graph of a quarter circle with angles of 0, 30, 45, 60, and 90 degrees inscribed. Equivalence of angles in radians shown.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137425870\" class=\"bc-section section\">\n<h4>Using a Calculator to Find Sine and Cosine<\/h4>\n<p id=\"fs-id1165137425875\">To find the cosine and sine of angles other than the <span class=\"no-emphasis\">special angles<\/span>, we turn to a computer or calculator. <strong>Be aware<\/strong>: Most calculators can be set into \u201cdegree\u201d or \u201cradian\u201d mode, which tells the calculator the units for the input value. When we evaluate [latex]\\mathrm{cos}\\left(30\\right)[\/latex] on our calculator, it will evaluate it as the cosine of 30 degrees if the calculator is in degree mode, or the cosine of 30 radians if the calculator is in radian mode.<\/p>\n<div id=\"fs-id1165137892278\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135415735\"><strong>Given an angle in radians, use a graphing calculator to find the cosine. <\/strong><\/p>\n<ol id=\"fs-id1165137534102\" type=\"1\">\n<li>If the calculator has degree mode and radian mode, set it to radian mode.<\/li>\n<li>Press the COS key.<\/li>\n<li>Enter the radian value of the angle and press the close-parentheses key &#8220;)&#8221;.<\/li>\n<li>Press ENTER.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_02_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137827676\">\n<div id=\"fs-id1165137812122\">\n<h3>Example 4:\u00a0 Using a Graphing Calculator to Find Sine and Cosine<\/h3>\n<p id=\"fs-id1165135439982\">Evaluate [latex]\\mathrm{cos}\\left(5.1\\right)[\/latex] using a graphing calculator or computer.<\/p>\n<\/div>\n<div id=\"fs-id1165135436451\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135436451\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135436451\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165131968061\">\n<p id=\"fs-id1165137530715\">Make sure your calculator is in radian mode.\u00a0 Enter the following keystrokes:<\/p>\n<p style=\"text-align: center\">\u00a0[latex]\\text{COS }\\left(5.1\\right)\\text{ ENTER}[\/latex]<\/p>\n<div id=\"fs-id1165135190973\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\mathrm{cos}\\left(5.1\\right)\\approx0.37798[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135190240\" class=\"bc-section section\">\n<div class=\"textbox tryit\">\n<h3>Try It #4<\/h3>\n<div id=\"fs-id1165137416877\">\n<p id=\"fs-id1165137416878\">Evaluate [latex]\\mathrm{sin}\\left(2.3\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137733611\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137733611\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137733611\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137733612\">approximately 0.74571<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h3>Reference Angles<\/h3>\n<p id=\"fs-id1165135586345\">For any given angle in the first quadrant, there is an angle in the second quadrant with the same <em>y<\/em>-value and therefore the same sine value. Because the sine value is the <em>y<\/em>-coordinate on the unit circle, the other angle with the same sine will share the same <em>y<\/em>-value, but have the opposite <em>x<\/em>-value. Therefore, its cosine value will be the opposite of the first angle\u2019s cosine value.<\/p>\n<p id=\"fs-id1165137469869\">Likewise, there will be an angle in the fourth quadrant with the same <em>x<\/em>-value and therefore the same cosine as the original angle in the first quadrant. The angle with the same cosine will share the same <em>x<\/em>-value but will have the opposite <em>y<\/em>-value. Therefore, its sine value will be the opposite of the original angle\u2019s sine value.<\/p>\n<p id=\"fs-id1165137418799\">As shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_014\">Figure 11<\/a>, angle [latex]\\alpha[\/latex] has the same sine value as angle [latex]t;[\/latex] the cosine values are opposites. Angle [latex]\\beta[\/latex] has the same cosine value as angle [latex]t;[\/latex] the sine values are opposites.<\/p>\n<div id=\"fs-id1165137433119\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{array}{lll}\\mathrm{sin}\\left(t\\right)=\\text{ }\\mathrm{sin}\\left(\\alpha \\right)\\text{ }\\hfill & \\text{and}\\hfill & \\mathrm{cos}\\left(t\\right)=-\\mathrm{cos}\\left(\\alpha \\right)\\hfill \\\\ \\mathrm{sin}\\left(t\\right)=-\\mathrm{sin}\\left(\\beta \\right)\\hfill & \\text{and}\\hfill & \\mathrm{cos}\\left(t\\right)=\\text{ }\\mathrm{cos}\\left(\\beta \\right)\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"attachment_2037\" style=\"width: 750px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2037\" class=\"wp-image-2037\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/23153603\/3-3-reference-angle.png\" alt=\"\" width=\"740\" height=\"335\" \/><\/p>\n<p id=\"caption-attachment-2037\" class=\"wp-caption-text\">Figure 11<\/p>\n<\/div>\n<h4>Finding Reference Angles<\/h4>\n<p id=\"fs-id1165135191896\">An angle\u2019s <strong>reference angle<\/strong> is the measure of the smallest, positive, acute angle [latex]t[\/latex] formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants.<\/p>\n<div id=\"fs-id1165135517171\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p>An angle\u2019s <strong>reference angle<\/strong> is the size of the smallest acute angle, [latex]{t}^{\\prime },[\/latex] formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134258657\">We can see that reference angle is always an angle between [latex]0^{\\circ}[\/latex] and [latex]90^{\\circ},[\/latex] or [latex]0[\/latex] and [latex]\\frac{\\pi }{2}[\/latex] radians. As we can see from <a class=\"autogenerated-content\" href=\"#Figure_05_02_020\">Figure 12<\/a>, for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.<\/p>\n<div id=\"Figure_05_02_020\" class=\"wp-caption aligncenter\">\n<div style=\"width: 752px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132303\/CNX_Precalc_Figure_05_01_019-1.jpg\" alt=\"Four side by side graphs. First graph shows an angle of t in quadrant 1 in it's normal position. Second graph shows an angle of t in quadrant 2 due to a rotation of pi minus t. Third graph shows an angle of t in quadrant 3 due to a rotation of t minus pi. Fourth graph shows an angle of t in quadrant 4 due to a rotation of two pi minus t.\" width=\"742\" height=\"252\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 12:\u00a0<\/strong>\u00a0Reference Angles<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134122927\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165133103951\"><strong>Given an angle between [latex]0[\/latex] and [latex]2\\pi,[\/latex] or between [latex]0^{\\circ}[\/latex] and [latex]360^{\\circ},[\/latex] find its reference angle.<\/strong><\/p>\n<ol id=\"fs-id1165137696399\" type=\"1\">\n<li>An angle in the first quadrant is its own reference angle.<\/li>\n<li>For an angle in the second quadrant, the reference angle is [latex]\\pi-t[\/latex] or [latex]180^{\\circ}-t.[\/latex]<\/li>\n<li>For an angle in the third quadrant, the reference angle is\u00a0[latex]t-\\pi[\/latex] or [latex]t-180^{\\circ}.[\/latex]<\/li>\n<li>For an angle in the fourth quadrant, the reference angle is [latex]2\\pi -t[\/latex] or [latex]360^{\\circ}-t.[\/latex]<\/li>\n<\/ol>\n<p><strong>Given an angle less than [latex]0[\/latex] or greater than [latex]2\\pi,[\/latex] or less than [latex]0^{\\circ}[\/latex] or greater than [latex]360^{\\circ},[\/latex] find its reference angle.<\/strong><\/p>\n<ol id=\"fs-id1165137696399\" type=\"1\">\n<li>If an angle is less than [latex]0[\/latex] or greater than [latex]2\\pi,[\/latex] add or subtract [latex]2\\pi[\/latex] as many times as needed to find a coterminal angle between [latex]0[\/latex] and [latex]2\\pi .[\/latex]\u00a0 Similarly, if working in degrees add or subtract [latex]360^{\\circ}[\/latex] as many times as needed to find a coterminal angle between [latex]0^{\\circ}[\/latex] and [latex]360^{\\circ}.[\/latex]<\/li>\n<li>Once you have an angle\u00a0between [latex]0[\/latex] and [latex]2\\pi,[\/latex] or\u00a0between [latex]0^{\\circ}[\/latex] and [latex]360^{\\circ},[\/latex] follow steps 1 &#8211; 4 above.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_02_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137455785\">\n<div id=\"fs-id1165137455788\">\n<h3>Example 5:\u00a0 Finding a Reference Angle<\/h3>\n<p id=\"fs-id1165135436617\">Find the reference angle of [latex]225^{\\circ}[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_016\">Figure 13<\/a>.<\/p>\n<div id=\"Figure_05_02_016\" class=\"small\">\n<div style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132307\/CNX_Precalc_Figure_05_02_016.jpg\" alt=\"Graph of circle with 225 degree angle inscribed.\" width=\"370\" height=\"291\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 13:\u00a0<\/strong>\u00a0Graph of circle with 225 degree angle inscribed.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137535343\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137535343\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137535343\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137535345\" style=\"text-align: left\">Because [latex]225^{\\circ}[\/latex] is in the third quadrant, the reference angle is [latex]225^{\\circ} -180^{\\circ}=45^{\\circ}.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137761979\" class=\"precalculus tryit\">\n<h3>Try It # 5<\/h3>\n<div id=\"ti_05_02_05\">\n<div id=\"fs-id1165137598484\">\n<p id=\"fs-id1165137598486\">Find the reference angle of [latex]\\frac{5\\pi }{3}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137806750\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137806750\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137806750\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137806751\">[latex]\\frac{\\pi }{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135458607\" class=\"bc-section section\">\n<h4 id=\"fs-id1165135477487\"><span style=\"color: #6c64ad;font-size: 1em;font-weight: 600\">Using Reference Angles to Find Exact Values for Cosine and Sine<\/span><\/h4>\n<div id=\"fs-id1165134190698\" class=\"bc-section section\">\n<p id=\"fs-id1165134259243\">We can find the cosine and sine of any angle in any quadrant if we know the cosine or sine of its reference angle. The absolute values of the cosine and sine of an angle are the same as those of the reference angle. The sign depends on the quadrant of the original angle. The cosine will be positive or negative depending on the sign of the <em>x<\/em>-values in that quadrant. The sine will be positive or negative depending on the sign of the <em>y<\/em>-values in that quadrant.<\/p>\n<p>For an angle which has a special angle as the reference angle ([latex]\\frac{\\pi}{6}, \\frac{\\pi}{4},[\/latex] or [latex]\\frac{\\pi}{3},[\/latex]) we can produce exact value outputs for sine and cosine.<\/p>\n<div id=\"fs-id1165137401083\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165134047575\"><strong>Given an angle in standard position, find the reference angle, and the cosine and sine of the original angle. <\/strong><\/p>\n<ol id=\"fs-id1165137530558\" type=\"1\">\n<li>Measure the angle between the terminal side of the given angle and the horizontal axis. That is the reference angle.<\/li>\n<li>Determine the values of the cosine and sine of the reference angle.<\/li>\n<li>Give the cosine the same sign as the <em>x<\/em>-values in the quadrant of the original angle.<\/li>\n<li>Give the sine the same sign as the <em>y<\/em>-values in the quadrant of the original angle.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_02_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135521301\">\n<div id=\"fs-id1165131990665\">\n<h3>Example 6:\u00a0 Using Reference Angles to Find Sine and Cosine<\/h3>\n<ol>\n<li>Using a reference angle, find the exact values of [latex]\\mathrm{cos}\\left(150^{\\circ}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(150^{\\circ}\\right).[\/latex]<\/li>\n<li>Find angles between [latex]0^{\\circ}[\/latex] and [latex]360^{\\circ}[\/latex] which have the same exact values as [latex]\\mathrm{cos}\\left(150^{\\circ}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(150^{\\circ}\\right).[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135534994\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135534994\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li style=\"list-style-type: none\">\n<ol>\n<li style=\"text-align: left\">150\u00b0 is located in the second quadrant. The angle it makes with the x-axis is 180\u00b0 \u2212 150\u00b0 = 30\u00b0, so the reference angle is 30\u00b0.<br \/>\nThis tells us that 150\u00b0 has the same sine and cosine values as 30\u00b0, except for the sign. We know that [latex]\\mathrm{cos}\\left(30^{\\circ}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2},[\/latex] and [latex]\\mathrm{sin}\\left(30^{\\circ}\\right)=\\frac{1}{2}.[\/latex][latex]\\\\[\/latex]Since 150\u00b0 is in the second quadrant, the x-coordinate of the point on the circle is negative, so the cosine value is negative. The y-coordinate is positive, so the sine value is positive.[latex]\\\\[\/latex]<\/p>\n<div id=\"fs-id1165134417907\" class=\"unnumbered\" style=\"text-align: center\">\n<div id=\"fs-id1165134417905\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\mathrm{cos}\\left(150^{\\circ}\\right)=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div class=\"unnumbered\" style=\"text-align: center\">[latex]\\mathrm{sin}\\left(150^{\\circ}\\right)=\\frac{1}{2}[\/latex][latex]\\\\[\/latex]<\/div>\n<\/div>\n<\/li>\n<li>From part 1, we know that\u00a0150\u00b0 is in quadrant 2 and has a reference angle of\u00a030\u00b0.\u00a0 We also discussed that the cosine value in quadrant 2 is negative.\u00a0 The other quadrant where cosine is negative is quadrant 3, since the x coordinate of any point in quadrant 3 is negative.\u00a0 Therefore, we need an angle in quadrant 3 that has a reference angle of 30\u00b0.\u00a0[latex]\\\\[\/latex]Since we know that the reference angle in quadrant 3 is [latex]t-180^{\\circ}[\/latex] where t is an angle in standard position, the angle we want can be found by solving the equation\n<div id=\"fs-id1165134417908\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}30^{\\circ}&=t-180^{\\circ}\\\\t&=180^{\\circ}+30^{\\circ}\\\\t&= 210^{\\circ}.\\\\\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p>Therefore [latex]\\mathrm{cos}\\left(150^{\\circ}\\right)=\\mathrm{cos}\\left(210^{\\circ}\\right).[\/latex] [latex]\\\\[\/latex]<\/p>\n<div>We discussed in part 1 that the sine value in quadrant 2 is positive.\u00a0 The other quadrant where sine is positive is quadrant 1, since the y coordinate is positive in quadrant 1.<\/div>\n<p>Therefore,\u00a0[latex]\\mathrm{sin}\\left(150^{\\circ}\\right)=\\mathrm{sin}\\left(30^{\\circ}\\right).[\/latex][latex]\\\\[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It #6<\/h3>\n<ol>\n<li>Use the reference angle of [latex]315^{\\circ}[\/latex] to find the exact values for\u00a0 [latex]\\mathrm{cos}\\left(315^{\\circ}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(315^{\\circ}\\right).[\/latex]<\/li>\n<li>Find angles between [latex]0^{\\circ}[\/latex] and [latex]360^{\\circ}[\/latex] which have the same exact values as [latex]\\mathrm{cos}\\left(315\u00b0\\right)[\/latex] and [latex]\\mathrm{sin}\\left(315^{\\circ}\\right).[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135193309\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135193309\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165137647526\" type=\"a\">\n<li>The reference angle is 45\u00b0 and the angle is in the fourth quadrant.\u00a0 [latex]\\mathrm{cos}\\left(315^{\\circ}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\text{ }\\mathrm{sin}\\left(315^{\\circ}\\right)=\\frac{\u2013\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/li>\n<li>[latex]\\mathrm{cos}\\left(315^{\\circ}\\right)=\\mathrm{cos}\\left(45^{\\circ}\\right)\\\\\\mathrm{sin}\\left(315^{\\circ}\\right)=\\mathrm{sin}\\left(225^{\\circ}\\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Example 7:\u00a0 Using Reference Angles to Find Sine and Cosine<\/h3>\n<ol>\n<li>Using a reference angle, find the exact values of [latex]\\mathrm{cos}\\left(\\frac{5\\pi}{4}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(\\frac{5\\pi}{4}\\right).[\/latex]<\/li>\n<li>Find angles between [latex]0\\text{ and }2\\pi[\/latex] which have the same exact values as\u00a0[latex]\\mathrm{cos}\\left(\\frac{5\\pi}{4}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(\\frac{5\\pi}{4}\\right).[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135534996\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135534996\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li style=\"text-align: left\">[latex]\\frac{5\\pi }{4}[\/latex] is in the third quadrant. Its reference angle is [latex]\\frac{5\\pi }{4}-\\pi =\\frac{\\pi }{4}.[\/latex]\u00a0 The cosine and sine of [latex]\\frac{\\pi }{4}[\/latex] are both [latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}.[\/latex]\u00a0 In the third quadrant, both [latex]x[\/latex] and [latex]y[\/latex] are negative, so:[latex]\\\\[\/latex]\n<p style=\"text-align: center\">[latex]\\mathrm{cos}\\left(\\frac{5\\pi }{4}\\right)=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\\\[\/latex][latex]\\mathrm{sin}\\left(\\frac{5\\pi }{4}\\right)=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/p>\n<\/li>\n<li>\u00a0We know from part 3 that\u00a0[latex]\\frac{5\\pi }{4}[\/latex] is in the third quadrant and its reference angle is [latex]\\frac{5\\pi }{4}-\\pi =\\frac{\\pi }{4}.[\/latex]\u00a0 We also know that both cosine and sine are negative in quadrant 3.\u00a0 Cosine is also negative in quadrant 2 since the [latex]x[\/latex]-coordinate is negative in quadrant 2, so we can use the fact that in quadrant 2, the reference angle is:\n<div id=\"fs-id1165134417904\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\\\[\/latex][latex]\\begin{align*}\\frac{\\pi }{4}&=\\pi-t && \\text{t is an angle in standard position. } \\\\t&=\\pi-\\frac{\\pi }{4}\\\\t&=\\frac{3\\pi}{4}\\end{align*}[\/latex]\u00a0[latex]\\\\[\/latex]<\/div>\n<p>We therefore know that [latex]\\mathrm{cos}\\left(\\frac{5\\pi }{4}\\right)=\\mathrm{cos}\\left(\\frac{3\\pi }{4}\\right).[\/latex][latex]\\\\[\/latex] Sine is also negative in quadrant 4 since the [latex]y[\/latex]-coordinate is negative in quadrant 4.\u00a0 We know that in quadrant 4 the reference angle is:<\/p>\n<div id=\"fs-id1165134417906\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}\\frac{\\pi}{4}&=2\\pi-t&&\\text{Now solve for t.}\\\\t&=2\\pi-\\frac{\\pi}{4}\\\\t&=\\frac{7\\pi}{4}\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p>We therefore know that [latex]\\mathrm{sin}\\left(\\frac{5\\pi}{4}\\right)=\\mathrm{sin}\\left(\\frac{7\\pi}{4}\\right).[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"fs-id1165137550671\" class=\"precalculus tryit\">\n<h3>Try It # 7<\/h3>\n<div id=\"ti_05_02_06\">\n<div id=\"fs-id1165134081332\">\n<ol id=\"fs-id1165134081334\" type=\"a\">\n<li>Use the reference angle of [latex]-\\frac{\\pi }{6}[\/latex] to find the exact values of\u00a0 [latex]\\mathrm{cos}\\left(-\\frac{\\pi}{6}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(-\\frac{\\pi}{6}\\right).[\/latex]<\/li>\n<li>Find angles between [latex]0\\text{ and }2\\pi[\/latex] which have the same exact values as\u00a0[latex]\\mathrm{cos}\\left(-\\frac{\\pi}{6}\\right)[\/latex] and [latex]\\mathrm{sin}\\left(-\\frac{\\pi}{6}\\right).[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135193308\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135193308\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135193308\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165137647526\" type=\"a\">\n<li>The reference angle is [latex]\\frac{\\pi}{6}[\/latex] and the angle is in the fourth quadrant. [latex]\\mathrm{cos}\\left(-\\frac{\\pi }{6}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2},\\text{ }\\mathrm{sin}\\left(-\\frac{\\pi }{6}\\right)=-\\frac{1}{2}[\/latex]<\/li>\n<li>[latex]\\mathrm{cos}\\left(-\\frac{\\pi }{6}\\right)=\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)=\\mathrm{cos}\\left(\\frac{11\\pi }{6}\\right).[\/latex][latex]\\\\[\/latex][latex]\\mathrm{sin}\\left(-\\frac{\\pi}{6}\\right)=\\mathrm{sin}\\left(\\frac{7\\pi}{6}\\right)=\\mathrm{cos}\\left(\\frac{11\\pi }{6}\\right).[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137766908\" class=\"bc-section section\">\n<h4>Using Reference Angles to Find Coordinates<\/h4>\n<p id=\"fs-id1165137766914\">Now that we have learned how to find the cosine and sine values for angles whose reference angles are acute special angles, the rest of the special angles on the unit circle can be determine. They are shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_017\">Figure 14<\/a>. Take time to learn the [latex]\\left(x,y\\right)[\/latex] coordinates of all of the major angles in the first quadrant.<\/p>\n<div id=\"Figure_05_02_017\" class=\"wp-caption aligncenter\">\n<div class=\"wp-caption-text\"><\/div>\n<div style=\"width: 749px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132311\/CNX_Precalc_Figure_05_02_017.jpg\" alt=\"Graph of unit circle with angles in degrees, angles in radians, and points along the circle inscribed.\" width=\"739\" height=\"478\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 14:<\/strong> Special angles and coordinates of corresponding points on the unit circle<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135704914\">In addition to learning the values for special angles, we can use reference angles to find [latex]\\left(x,y\\right)[\/latex] coordinates of any point on the unit circle, using what we know of reference angles along with the <span class=\"no-emphasis\">identities<\/span><\/p>\n<div id=\"eip-150\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}x&=\\mathrm{cos}\\left(t\\right) \\\\ y&=\\mathrm{sin}\\left(t\\right).\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137411099\">First we find the reference angle corresponding to the given angle. Then we take the sine and cosine values of the <span class=\"no-emphasis\">reference angle<\/span>, and give them the signs corresponding to the <em>y<\/em>&#8211; and <em>x<\/em>-values of the quadrant.<\/p>\n<div id=\"fs-id1165137676498\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135381357\"><strong>Given the angle of a point on a unit circle, find the [latex]\\left(x,y\\right)[\/latex] coordinates of the point.<\/strong><\/p>\n<ol id=\"fs-id1165134211369\" type=\"1\">\n<li>Find the reference angle by measuring the smallest angle to the <em>x<\/em>-axis.<\/li>\n<li>Find the cosine and sine of the reference angle.<\/li>\n<li>Determine the appropriate signs for [latex]x[\/latex] and [latex]y[\/latex] in the given quadrant.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_02_07\" class=\"textbox examples\">\n<div id=\"fs-id1165137409939\">\n<div id=\"fs-id1165137409941\">\n<h3>Example 8:\u00a0 Using the Unit Circle to Find Coordinates<\/h3>\n<p id=\"fs-id1165137409947\">Find the coordinates of the point on the unit circle at an angle of [latex]\\frac{7\\pi }{6}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137704533\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137704533\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137704533\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137704536\">We know that the angle [latex]\\frac{7\\pi }{6}[\/latex] is in the third quadrant.<\/p>\n<p id=\"fs-id1165137851856\">First, let\u2019s find the reference angle by measuring the angle to the <em>x<\/em>-axis. To find the reference angle of an angle whose terminal side is in quadrant III, we find the difference of the angle and [latex]\\pi .[\/latex]<\/p>\n<div id=\"fs-id1165134555590\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\frac{7\\pi }{6}-\\pi =\\frac{\\pi }{6}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137394109\">Next, we will find the cosine and sine of the reference angle:<\/p>\n<div id=\"fs-id1165137394112\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)=\\frac{1}{2}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165132011271\">We must determine the appropriate signs for <em>x<\/em> and <em>y<\/em> in the given quadrant. Because our original angle is in the third quadrant, where both [latex]x[\/latex] and [latex]y[\/latex] are negative, both cosine and sine are negative.<\/p>\n<div id=\"fs-id1165137758918\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}\\mathrm{cos}\\left(\\frac{7\\pi }{6}\\right)&=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2} \\\\ \\mathrm{sin}\\left(\\frac{7\\pi }{6}\\right)&=-\\frac{1}{2}\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135503862\">Now we can calculate the [latex]\\left(x,y\\right)[\/latex] coordinates using the identities [latex]x=\\mathrm{cos}\\left(\\theta\\right)[\/latex] and [latex]y=\\mathrm{sin}\\left(\\theta\\right) .[\/latex]<\/p>\n<p id=\"fs-id1165137411025\">The coordinates of the point are [latex]\\left(-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2},-\\frac{1}{2}\\right)[\/latex] on the unit circle.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137407555\" class=\"precalculus tryit\">\n<h3>Try It # 8<\/h3>\n<div id=\"ti_05_02_07\">\n<div id=\"fs-id1165137535595\">\n<p id=\"fs-id1165137535596\">Find the coordinates of the point on the unit circle at an angle of [latex]\\frac{5\\pi }{3}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135530358\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135530358\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135530358\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135530359\">[latex]\\left(\\frac{1}{2},-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137583095\" class=\"precalculus media\">\n<h3>Circles with Radius Different Than 1<\/h3>\n<p>Suppose we have a circle centered at the origin other than the unit circle. How can we find the coordinates of a point where the terminal side of the angle in standard position intersects the circle? We know from Section 3.1 that the sine and cosine values of an acute angle do not change regardless of the size of the triangle. This means that [latex]\\mathrm{cos}\\left(t\\right)[\/latex] will be the same whether [latex]t[\/latex] is in a triangle with small or large sides. However, it is clear that the coordinates of the point of intersection of the terminal side with the circle of radius r will not be the same as the coordinates of the point on the unit circle.<\/p>\n<p>We can apply similar reasoning to considering values of sine and cosine values on a circle that is not a unit circle as we did earlier in this section. By drawing an angle in standard position in quadrant 1, we can drop a perpendicular from the point of intersection of the terminal side with the circle whose center is at the origin to the x axis. If the point on the circle is (x, y), then the sides of the right triangle formed by dropping the perpendicular will also be [latex]x\\text{ and }y.[\/latex] Now, instead of the radius being 1, we would have a radius of [latex]r.[\/latex] Therefore, [latex]\\mathrm{cos}\\left(t\\right)=\\frac{x}{r}\\text{ and }\\mathrm{sin}\\left(t\\right)=\\frac{y}{r}.[\/latex] This means that [latex]x=r\\mathrm{cos}\\left(t\\right)\\text{ and }y=r\\mathrm{sin}\\left(t\\right).[\/latex]<\/p>\n<div class=\"textbox Definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165137793724\">A point [latex]\\left(x,y\\right)[\/latex] on the circle, centered at the origin, with radius [latex]r[\/latex] corresponds to<\/p>\n<div id=\"Equation_05_02_01\" style=\"text-align: center\">[latex]\\begin{align*}x&=r\\mathrm{cos}\\left(t\\right)\\text{ and }\\\\y&=r\\mathrm{sin}\\left(t\\right)\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<div>where [latex]t[\/latex] is the angle in standard position.<\/div>\n<\/div>\n<div id=\"Equation_05_02_02\">\n<div id=\"fs-id1165137704533\">\n<div class=\"textbox examples\">\n<h3>Example 9: \u00a0Finding Coordinates on a Circle with Radius [latex]r[\/latex]<\/h3>\n<p>Find the coordinates of the point on the circle with radius 5 at an angle of [latex]\\frac{7\\pi}{6}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137409939\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137409939\" class=\"hidden-answer\" style=\"display: none\">\nWe already know from Example 7 that:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align*}\\mathrm{cos}\\left(\\frac{7\\pi }{6}\\right)&=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2} \\\\ \\mathrm{sin}\\left(\\frac{7\\pi }{6}\\right)&=-\\frac{1}{2}\\end{align*}[\/latex][latex]\\\\[\/latex]<\/p>\n<p id=\"fs-id1165135503862\" style=\"text-align: left\">Now we can calculate the [latex]\\left(x,y\\right)[\/latex] coordinates using the identities<\/p>\n<p style=\"text-align: center\">[latex]x=r\\mathrm{cos}\\left(\\theta\\right)[\/latex] and [latex]y=r\\mathrm{sin}\\left(\\theta\\right) .[\/latex]<\/p>\n<p id=\"fs-id1165137411025\">The coordinates of the point are [latex]\\left(-\\frac{5\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2},-\\frac{5}{2}\\right)[\/latex] on the circle with radius 5.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137407555\" class=\"precalculus tryit\">\n<h3>Try It # 9<\/h3>\n<div id=\"ti_05_02_07\">\n<div id=\"fs-id1165137535595\">\n<p id=\"fs-id1165137535596\">Find the coordinates of the point on the circle with radius 7 at an angle of [latex]\\frac{5\\pi }{3}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135530360\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135530360\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135530359\">[latex]\\left(\\frac{7}{2},-\\frac{7\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h3>Even and Odd Functions<\/h3>\n<p id=\"fs-id1165137571788\">To be able to use our sine and cosine functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.<\/p>\n<p>Recall that:<\/p>\n<ul>\n<li id=\"fs-id1165134042164\">An <span class=\"no-emphasis\">even function<\/span> is one in which [latex]f\\left(-x\\right)=f\\left(x\\right).[\/latex]<\/li>\n<li id=\"fs-id1165137443964\">An <span class=\"no-emphasis\">odd function<\/span> is one in which [latex]f\\left(-x\\right)=-f\\left(x\\right).[\/latex]<\/li>\n<\/ul>\n<p>We can test whether a trigonometric function is even or odd by drawing a <span class=\"no-emphasis\">unit circle<\/span> with a positive and a negative angle, as in <a class=\"autogenerated-content\" href=\"#Figure_05_03_007\">\u00a0Figure 15.\u00a0<\/a>The sine of the positive angle is [latex]y.[\/latex] The sine of the negative angle is [latex]-y.[\/latex]The <span class=\"no-emphasis\">sine function<\/span>, then, is an odd function. \u00a0The cosine of the positive angle is [latex]x[\/latex], as is the cosine of the negative angle. \u00a0Therefore, the cosine function is an even function.<\/p>\n<div id=\"Figure_05_03_007\" class=\"small\">\n<div style=\"width: 360px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132429\/CNX_Precalc_Figure_05_03_007.jpg\" alt=\"Graph of circle with angle of t and -t inscribed. Point of (x, y) is at intersection of terminal side of angle t and edge of circle. Point of (x, -y) is at intersection of terminal side of angle -t and edge of circle.\" width=\"350\" height=\"265\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 15<\/p>\n<\/div>\n<\/div>\n<p>We can summarize this by saying:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align*}\\mathrm{sin}\\left(-x\\right)&=-\\mathrm{sin}\\left(x\\right)\\\\ \\mathrm{cos}\\left(-x\\right)&=\\mathrm{cos}\\left(x\\right)\\end{align*}[\/latex]<\/p>\n<p id=\"fs-id1165137828452\">Access these online resources for additional instruction and practice with sine and cosine functions.<\/p>\n<ul id=\"fs-id1165135160446\">\n<li><a href=\"http:\/\/openstax.org\/l\/trigunitcir\">Trigonometric Functions Using the Unit Circle<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/sincosuc\">Sine and Cosine from the Unit Circle<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/sincosmult\">Sine and Cosine from the Unit Circle and Multiples of Pi Divided by Six<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/sincosmult4\">Sine and Cosine from the Unit Circle and Multiples of Pi Divided by Four<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/trigrefang\">Trigonometric Functions Using Reference Angles<\/a><\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165133238450\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165134284283\" summary=\"..\">\n<tbody>\n<tr>\n<td class=\"border\">Cosine<\/td>\n<td class=\"border\">[latex]\\mathrm{cos}\\left(t\\right)=x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Sine<\/td>\n<td class=\"border\">[latex]\\mathrm{sin}\\left(t\\right)=y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Pythagorean Identity<\/td>\n<td class=\"border\">[latex]{\\mathrm{cos}}^{2}\\left(t\\right)+{\\mathrm{sin}}^{2}\\left(t\\right)=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137692629\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137772132\">\n<li>Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of 1 unit.<\/li>\n<li>Using the unit circle, the sine of an angle [latex]t[\/latex] equals the <em>y<\/em>-value of the endpoint on the unit circle of an arc of length [latex]t[\/latex] whereas the cosine of an angle [latex]t[\/latex] equals the <em>x<\/em>-value of the endpoint.<\/li>\n<li>When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles.<\/li>\n<li>Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known.<\/li>\n<li>The domain of the sine and cosine functions is all real numbers.<\/li>\n<li>The range of both the sine and cosine functions is [latex]\\left[-1,1\\right].[\/latex]<\/li>\n<li>The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.<\/li>\n<li>The signs of the sine and cosine are determined from the <em>x<\/em>&#8211; and <em>y<\/em>-values in the quadrant of the original angle.<\/li>\n<li>An angle\u2019s reference angle is the size angle, [latex]t,[\/latex] formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis.<\/li>\n<li>Reference angles can be used to find the sine and cosine of the original angle.<\/li>\n<li>Reference angles can also be used to find the coordinates of a point on a unit circle.<\/li>\n<li>When the radius of a circle centered at the origin is not 1, we can find coordinates of a point on the circle by multiplying the sine and cosine of the angle by [latex]r.[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137762325\">\n<dt>cosine function<\/dt>\n<dd id=\"fs-id1165137408620\">the <em>x<\/em>-value of the point on a unit circle corresponding to a given angle<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137812230\">\n<dt>Pythagorean Identity<\/dt>\n<dd id=\"fs-id1165137812235\">a corollary of the Pythagorean Theorem stating that the square of the cosine of a given angle plus the square of the sine of that angle equals 1<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137812240\">\n<dt>sine function<\/dt>\n<dd id=\"fs-id1165137749964\">the <em>y<\/em>-value of the point on a unit circle corresponding to a given angle<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137749972\">\n<dt>unit circle<\/dt>\n<dd id=\"fs-id1165135693774\">a circle with a center at [latex]\\left(0,0\\right)[\/latex] and radius 1.<\/dd>\n<\/dl>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-557\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit Circle: Sine and Cosine Functions. <strong>Authored by<\/strong>: Douglas Hoffman. <strong>Provided by<\/strong>: Openstax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:F6bX9ckM@7\/Unit-Circle-Sine-and-Cosine-Functions\">https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:F6bX9ckM@7\/Unit-Circle-Sine-and-Cosine-Functions<\/a>. <strong>Project<\/strong>: Essential Precalcus, Part 2. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":311,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Unit Circle: Sine and Cosine Functions\",\"author\":\"Douglas Hoffman\",\"organization\":\"Openstax\",\"url\":\"https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:F6bX9ckM@7\/Unit-Circle-Sine-and-Cosine-Functions\",\"project\":\"Essential Precalcus, Part 2\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-557","chapter","type-chapter","status-publish","hentry"],"part":478,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/557","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":69,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/557\/revisions"}],"predecessor-version":[{"id":3244,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/557\/revisions\/3244"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/478"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/557\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=557"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=557"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=557"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=557"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}