{"id":193,"date":"2021-06-04T18:09:56","date_gmt":"2021-06-04T18:09:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/chapter\/sum-and-difference-identities\/"},"modified":"2021-06-14T23:50:14","modified_gmt":"2021-06-14T23:50:14","slug":"sum-and-difference-identities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/chapter\/sum-and-difference-identities\/","title":{"raw":"4.2 Sum and Difference Identities","rendered":"4.2 Sum and Difference Identities"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Use sum and difference formulas for cosine.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use sum and difference formulas for sine.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use sum and difference formulas for tangent.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use sum and difference formulas for cofunctions.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use sum and difference formulas to verify identities.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Use sum and difference formulas for cosine<\/h2>\r\nFinding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values. We can use the <strong>special angles<\/strong>, which we can review in the unit circle shown in Figure 2.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164036\/CNX_Precalc_Figure_07_01_0042.jpg\" alt=\"Diagram of the unit circle with points labeled on its edge. P point is at an angle a from the positive x axis with coordinates (cosa, sina). Point Q is at an angle of B from the positive x axis with coordinates (cosb, sinb). Angle POQ is a - B degrees. Point A is at an angle of (a-B) from the x axis with coordinates (cos(a-B), sin(a-B)). Point B is just at point (1,0). Angle AOB is also a - B degrees. Radii PO, AO, QO, and BO are all 1 unit long and are the legs of triangles POQ and AOB. Triangle POQ is a rotation of triangle AOB, so the distance from P to Q is the same as the distance from A to B. \" width=\"975\" height=\"638\" \/> <b>Figure 2.<\/b> The Unit Circle[\/caption]\r\n\r\nWe will begin with the <strong>sum and difference formulas for cosine<\/strong>, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles.\r\n<table id=\"Table_07_02_01\" summary=\"Two rows, two columns. The table has ordered pairs of these row values: (Sum formula for cosine, cos(a+B) = cos(a)cos(B) - sin(a)sin(B)) and (Difference formula for cosine, cos(a-B) = cos(a)cos(B) + sin(a)sin(B)).\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Sum formula for cosine<\/strong><\/td>\r\n<td>[latex]\\cos \\left(\\alpha +\\beta \\right)=\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Difference formula for cosine<\/strong><\/td>\r\n<td>[latex]\\cos \\left(\\alpha -\\beta \\right)=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFirst, we will prove the difference formula for cosines. Let\u2019s consider two points on the unit circle. Point [latex]P[\/latex] is at an angle [latex]\\alpha [\/latex] from the positive <em>x-<\/em>axis with coordinates [latex]\\left(\\cos \\alpha ,\\sin \\alpha \\right)[\/latex] and point [latex]Q[\/latex] is at an angle of [latex]\\beta [\/latex] from the positive <em>x-<\/em>axis with coordinates [latex]\\left(\\cos \\beta ,\\sin \\beta \\right)[\/latex]. Note the measure of angle [latex]POQ[\/latex] is [latex]\\alpha -\\beta [\/latex].\r\n\r\nLabel two more points: [latex]A[\/latex] at an angle of [latex]\\left(\\alpha -\\beta \\right)[\/latex] from the positive <em>x-<\/em>axis with coordinates [latex]\\left(\\cos \\left(\\alpha -\\beta \\right),\\sin \\left(\\alpha -\\beta \\right)\\right)[\/latex]; and point [latex]B[\/latex] with coordinates [latex]\\left(1,0\\right)[\/latex]. Triangle [latex]POQ[\/latex] is a rotation of triangle [latex]AOB[\/latex] and thus the distance from [latex]P[\/latex] to [latex]Q[\/latex] is the same as the distance from [latex]A[\/latex] to [latex]B[\/latex].\r\n<figure id=\"Figure_07_02_002\" class=\"small\"><span id=\"fs-id1130636\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164038\/CNX_Precalc_Figure_07_02_0022.jpg\" alt=\"Diagram of the unit circle with points labeled on its edge. P point is at an angle a from the positive x axis with coordinates (cosa, sina). Point Q is at an angle of B from the positive x axis with coordinates (cosb, sinb). Angle POQ is a - B degrees. Point A is at an angle of (a-B) from the x axis with coordinates (cos(a-B), sin(a-B)). Point B is just at point (1,0). Angle AOB is also a - B degrees. Radii PO, AO, QO, and BO are all 1 unit long and are the legs of triangles POQ and AOB. Triangle POQ is a rotation of triangle AOB, so the distance from P to Q is the same as the distance from A to B.\" \/><\/span><\/figure>\r\n<p style=\"text-align: center;\"><strong>Figure 3.\u00a0<\/strong>We can find the distance from [latex]P[\/latex] to [latex]Q[\/latex] using the <strong>distance formula<\/strong>.<\/p>\r\n\r\n<div><\/div>\r\n<div style=\"text-align: center;\">[latex]\\begin{align}{d}_{PQ}&amp;=\\sqrt{{\\left(\\cos \\alpha -\\cos \\beta \\right)}^{2}+{\\left(\\sin \\alpha -\\sin \\beta \\right)}^{2}} \\\\ &amp;=\\sqrt{{\\cos }^{2}\\alpha -2\\cos \\alpha \\cos \\beta +{\\cos }^{2}\\beta +{\\sin }^{2}\\alpha -2\\sin \\alpha \\sin \\beta +{\\sin }^{2}\\beta } \\end{align}[\/latex]<\/div>\r\nThen we apply the <strong>Pythagorean identity<\/strong> and simplify.\r\n<div style=\"text-align: center;\">[latex]\\begin{align} &amp;=\\sqrt{\\left({\\cos }^{2}\\alpha +{\\sin }^{2}\\alpha \\right)+\\left({\\cos }^{2}\\beta +{\\sin }^{2}\\beta \\right)-2\\cos \\alpha \\cos \\beta -2\\sin \\alpha \\sin \\beta } \\\\ &amp;=\\sqrt{1+1 - 2\\cos \\alpha \\cos \\beta -2\\sin \\alpha \\sin \\beta } \\\\ &amp;=\\sqrt{2 - 2\\cos \\alpha \\cos \\beta -2\\sin \\alpha \\sin \\beta } \\end{align}[\/latex]<\/div>\r\nSimilarly, using the distance formula we can find the distance from [latex]A[\/latex] to [latex]B[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}{d}_{AB}&amp;=\\sqrt{{\\left(\\cos \\left(\\alpha -\\beta \\right)-1\\right)}^{2}+{\\left(\\sin \\left(\\alpha -\\beta \\right)-0\\right)}^{2}} \\\\ &amp;=\\sqrt{{\\cos }^{2}\\left(\\alpha -\\beta \\right)-2\\cos \\left(\\alpha -\\beta \\right)+1+{\\sin }^{2}\\left(\\alpha -\\beta \\right)} \\end{align}[\/latex]<\/div>\r\nApplying the Pythagorean identity and simplifying we get:\r\n<div style=\"text-align: center;\">[latex]\\begin{align} &amp;=\\sqrt{\\left({\\cos }^{2}\\left(\\alpha -\\beta \\right)+{\\sin }^{2}\\left(\\alpha -\\beta \\right)\\right)-2\\cos \\left(\\alpha -\\beta \\right)+1}\\\\ &amp;=\\sqrt{1 - 2\\cos \\left(\\alpha -\\beta \\right)+1}\\\\ &amp;=\\sqrt{2 - 2\\cos \\left(\\alpha -\\beta \\right)} \\end{align}[\/latex]<\/div>\r\nBecause the two distances are the same, we set them equal to each other and simplify.\r\n<div style=\"text-align: center;\">[latex]\\begin{align} \\sqrt{2 - 2\\cos \\alpha \\cos \\beta -2\\sin \\alpha \\sin \\beta }&amp;=\\sqrt{2 - 2\\cos \\left(\\alpha -\\beta \\right)} \\\\ 2 - 2\\cos \\alpha \\cos \\beta -2\\sin \\alpha \\sin \\beta &amp;=2 - 2\\cos \\left(\\alpha -\\beta \\right) \\end{align}[\/latex]<\/div>\r\nFinally we subtract [latex]2[\/latex] from both sides and divide both sides by [latex]-2[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta =\\cos \\left(\\alpha -\\beta \\right)\\end{align}[\/latex]<\/div>\r\nThus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Sum and Difference Formulas for Cosine<\/h3>\r\nThese formulas can be used to calculate the cosine of sums and differences of angles.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha +\\beta \\right)=\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta\\end{align} [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha -\\beta \\right)=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta\\end{align} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given two angles, find the cosine of the difference between the angles.<strong>\r\n<\/strong><\/h3>\r\n<ol id=\"fs-id1374215\">\r\n \t<li>Write the difference formula for cosine.<\/li>\r\n \t<li>Substitute the values of the given angles into the formula.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 1: Finding the Exact Value Using the Formula for the Cosine of the Difference of Two Angles<\/h3>\r\nUsing the formula for the cosine of the difference of two angles, find the exact value of [latex]\\cos \\left(\\frac{5\\pi }{4}-\\frac{\\pi }{6}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"956915\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"956915\"]\r\n\r\nUse the formula for the cosine of the difference of two angles. We have\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha -\\beta \\right)&amp;=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta \\\\ \\cos \\left(\\frac{5\\pi }{4}-\\frac{\\pi }{6}\\right)&amp;=\\cos \\left(\\frac{5\\pi }{4}\\right)\\cos \\left(\\frac{\\pi }{6}\\right)+\\sin \\left(\\frac{5\\pi }{4}\\right)\\sin \\left(\\frac{\\pi }{6}\\right) \\\\ &amp;=\\left(-\\frac{\\sqrt{2}}{2}\\right)\\left(\\frac{\\sqrt{3}}{2}\\right)-\\left(\\frac{\\sqrt{2}}{2}\\right)\\left(\\frac{1}{2}\\right) \\\\ &amp;=-\\frac{\\sqrt{6}}{4}-\\frac{\\sqrt{2}}{4} \\\\ &amp;=\\frac{-\\sqrt{6}-\\sqrt{2}}{4} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nFind the exact value of [latex]\\cos \\left(\\frac{\\pi }{3}-\\frac{\\pi }{4}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"700317\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"700317\"]\r\n\r\n[latex]\\frac{\\sqrt{2}+\\sqrt{6}}{4}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 2: Finding the Exact Value Using the Formula for the Sum of Two Angles for Cosine<\/h3>\r\nFind the exact value of [latex]\\cos \\left({75}^{\\circ }\\right)[\/latex].\r\n\r\n[reveal-answer q=\"617422\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"617422\"]\r\n\r\nAs [latex]{75}^{\\circ }={45}^{\\circ }+{30}^{\\circ }[\/latex], we can evaluate [latex]\\cos \\left({75}^{\\circ }\\right)[\/latex] as [latex]\\cos \\left({45}^{\\circ }+{30}^{\\circ }\\right)[\/latex]. Thus,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left({45}^{\\circ }+{30}^{\\circ }\\right)&amp;=\\cos \\left({45}^{\\circ }\\right)\\cos \\left({30}^{\\circ }\\right)-\\sin \\left({45}^{\\circ }\\right)\\sin \\left({30}^{\\circ }\\right) \\\\ &amp;=\\frac{\\sqrt{2}}{2}\\left(\\frac{\\sqrt{3}}{2}\\right)-\\frac{\\sqrt{2}}{2}\\left(\\frac{1}{2}\\right) \\\\ &amp;=\\frac{\\sqrt{6}}{4}-\\frac{\\sqrt{2}}{4} \\\\ &amp;=\\frac{\\sqrt{6}-\\sqrt{2}}{4} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nFind the exact value of [latex]\\cos \\left({105}^{\\circ }\\right)[\/latex].\r\n\r\n[reveal-answer q=\"683323\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"683323\"]\r\n\r\n[latex]\\frac{\\sqrt{2}-\\sqrt{6}}{4}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]173445[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use sum and difference formulas for sine<\/h2>\r\nThe <strong>sum and difference formulas for sine<\/strong> can be derived in the same manner as those for cosine, and they resemble the cosine formulas.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Sum and Difference Formulas for Sine<\/h3>\r\nThese formulas can be used to calculate the sines of sums and differences of angles.\r\n<p style=\"text-align: center;\">[latex]\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta [\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given two angles, find the sine of the difference between the angles.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Write the difference formula for sine.<\/li>\r\n \t<li>Substitute the given angles into the formula.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 3: Using Sum and Difference Identities to Evaluate the Difference of Angles<\/h3>\r\nUse the sum and difference identities to evaluate the difference of the angles and show that part <em>a<\/em> equals part <em>b.<\/em>\r\n<ol>\r\n \t<li>[latex]\\sin \\left({45}^{\\circ }-{30}^{\\circ }\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\sin \\left({135}^{\\circ }-{120}^{\\circ }\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"191680\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"191680\"]\r\n<ol>\r\n \t<li>Let\u2019s begin by writing the formula and substitute the given angles.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left(\\alpha -\\beta \\right)&amp;=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta \\\\ \\sin \\left({45}^{\\circ }-{30}^{\\circ }\\right)&amp;=\\sin \\left({45}^{\\circ }\\right)\\cos \\left({30}^{\\circ }\\right)-\\cos \\left({45}^{\\circ }\\right)\\sin \\left({30}^{\\circ }\\right) \\end{align}[\/latex]<\/div>\r\nNext, we need to find the values of the trigonometric expressions.\r\n<div style=\"text-align: center;\">[latex]\\sin \\left({45}^{\\circ }\\right)=\\frac{\\sqrt{2}}{2},\\text{ }\\cos \\left({30}^{\\circ }\\right)=\\frac{\\sqrt{3}}{2},\\text{ }\\cos \\left({45}^{\\circ }\\right)=\\frac{\\sqrt{2}}{2},\\text{ }\\sin \\left({30}^{\\circ }\\right)=\\frac{1}{2}[\/latex]<\/div>\r\nNow we can substitute these values into the equation and simplify.\r\n<div style=\"text-align: center;\">[latex]\\begin{align} \\sin \\left({45}^{\\circ }-{30}^{\\circ }\\right)&amp;=\\frac{\\sqrt{2}}{2}\\left(\\frac{\\sqrt{3}}{2}\\right)-\\frac{\\sqrt{2}}{2}\\left(\\frac{1}{2}\\right) \\\\ &amp;=\\frac{\\sqrt{6}-\\sqrt{2}}{4}\\hfill \\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Again, we write the formula and substitute the given angles.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left(\\alpha -\\beta \\right)&amp;=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta\\\\ \\sin \\left({135}^{\\circ }-{120}^{\\circ }\\right)&amp;=\\sin \\left({135}^{\\circ }\\right)\\cos \\left({120}^{\\circ }\\right)-\\cos \\left({135}^{\\circ }\\right)\\sin \\left({120}^{\\circ }\\right)\\end{align}[\/latex]<\/div>\r\nNext, we find the values of the trigonometric expressions.\r\n<div style=\"text-align: center;\">[latex]\\sin \\left({135}^{\\circ }\\right)=\\frac{\\sqrt{2}}{2},\\cos \\left({120}^{\\circ }\\right)=-\\frac{1}{2},\\cos \\left({135}^{\\circ }\\right)=\\frac{\\sqrt{2}}{2},\\sin \\left({120}^{\\circ }\\right)=\\frac{\\sqrt{3}}{2}[\/latex]<\/div>\r\nNow we can substitute these values into the equation and simplify.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left({135}^{\\circ }-{120}^{\\circ }\\right)&amp;=\\frac{\\sqrt{2}}{2}\\left(-\\frac{1}{2}\\right)-\\left(-\\frac{\\sqrt{2}}{2}\\right)\\left(\\frac{\\sqrt{3}}{2}\\right) \\\\ &amp;=\\frac{-\\sqrt{2}+\\sqrt{6}}{4} \\\\ &amp;=\\frac{\\sqrt{6}-\\sqrt{2}}{4} \\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 4: Finding the Exact Value of an Expression Involving an Inverse Trigonometric Function<\/h3>\r\nFind the exact value of [latex]\\sin \\left({\\cos }^{-1}\\frac{1}{2}+{\\sin }^{-1}\\frac{3}{5}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"954448\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"954448\"]\r\n\r\nThe pattern displayed in this problem is [latex]\\sin \\left(\\alpha +\\beta \\right)[\/latex]. Let [latex]\\alpha ={\\cos }^{-1}\\frac{1}{2}[\/latex] and [latex]\\beta ={\\sin }^{-1}\\frac{3}{5}[\/latex]. Then we can write\r\n<p style=\"text-align: center;\">[latex]\\begin{align} \\cos \\alpha &amp;=\\frac{1}{2},0\\le \\alpha \\le \\pi\\\\ \\sin \\beta &amp;=\\frac{3}{5},-\\frac{\\pi }{2}\\le \\beta \\le \\frac{\\pi }{2} \\end{align}[\/latex]<\/p>\r\nWe will use the Pythagorean identities to find [latex]\\sin \\alpha [\/latex] and [latex]\\cos \\beta [\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sin \\alpha &amp;=\\sqrt{1-{\\cos }^{2}\\alpha } \\\\ &amp;=\\sqrt{1-\\frac{1}{4}} \\\\ &amp;=\\sqrt{\\frac{3}{4}} \\\\ &amp;=\\frac{\\sqrt{3}}{2} \\\\ \\cos \\beta &amp;=\\sqrt{1-{\\sin }^{2}\\beta } \\\\ &amp;=\\sqrt{1-\\frac{9}{25}} \\\\ &amp;=\\sqrt{\\frac{16}{25}} \\\\ &amp;=\\frac{4}{5}\\end{align}[\/latex]<\/p>\r\nUsing the sum formula for sine,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left({\\cos }^{-1}\\frac{1}{2}+{\\sin }^{-1}\\frac{3}{5}\\right)&amp;=\\sin \\left(\\alpha +\\beta \\right) \\\\ &amp;=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta \\\\ &amp;=\\frac{\\sqrt{3}}{2}\\cdot \\frac{4}{5}+\\frac{1}{2}\\cdot \\frac{3}{5} \\\\ &amp;=\\frac{4\\sqrt{3}+3}{10}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]173443[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use sum and difference formulas for tangent<\/h2>\r\nFinding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.\r\n\r\nFinding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall, [latex]\\tan x=\\frac{\\sin x}{\\cos x},\\cos x\\ne 0[\/latex].\r\n\r\nLet\u2019s derive the sum formula for tangent.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\alpha +\\beta \\right)&amp;=\\frac{\\sin \\left(\\alpha +\\beta \\right)}{\\cos \\left(\\alpha +\\beta \\right)} \\\\[1mm] &amp;=\\frac{\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta } \\\\[1mm] &amp;=\\frac{\\frac{\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }}{\\frac{\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }} &amp;&amp; \\text{Divide the numerator and denominator by cos}\\alpha \\text{cos}\\beta \\\\[1mm] &amp;=\\frac{\\frac{\\sin \\alpha \\cos \\beta }{\\cos \\alpha \\cos \\beta}+\\frac{\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }}{\\frac{\\cos \\alpha \\cos \\beta }{\\cos \\alpha \\cos \\beta }-\\frac{\\sin \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }} &amp;&amp; \\text{Split the fractions.} \\\\[1mm] &amp;=\\frac{\\frac{\\sin \\alpha }{\\cos \\alpha }+\\frac{\\sin \\beta }{\\cos \\beta }}{1-\\frac{\\sin \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }} &amp;&amp; \\text{Cancel.} \\\\[1mm] &amp;=\\frac{\\tan \\alpha +\\tan \\beta }{1-\\tan \\alpha \\tan \\beta } \\end{align}[\/latex]<\/div>\r\nWe can derive the difference formula for tangent in a similar way.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Sum and Difference Formulas for Tangent<\/h3>\r\nThe <strong>sum and difference formulas for tangent<\/strong> are:\r\n<p style=\"text-align: center;\">[latex]\\tan \\left(\\alpha +\\beta \\right)=\\frac{\\tan \\alpha +\\tan \\beta }{1-\\tan \\alpha \\tan \\beta }[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\tan \\left(\\alpha -\\beta \\right)=\\frac{\\tan \\alpha -\\tan \\beta }{1+\\tan \\alpha \\tan \\beta }[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given two angles, find the tangent of the sum of the angles.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Write the sum formula for tangent.<\/li>\r\n \t<li>Substitute the given angles into the formula.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 5: Finding the Exact Value of an Expression Involving Tangent<\/h3>\r\nFind the exact value of [latex]\\tan \\left(\\frac{\\pi }{6}+\\frac{\\pi }{4}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"796417\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"796417\"]\r\n\r\nLet\u2019s first write the sum formula for tangent and substitute the given angles into the formula.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\alpha +\\beta \\right)&amp;=\\frac{\\tan \\alpha +\\tan \\beta }{1-\\tan \\alpha \\tan \\beta }\\\\ \\tan \\left(\\frac{\\pi }{6}+\\frac{\\pi }{4}\\right)&amp;=\\frac{\\tan \\left(\\frac{\\pi }{6}\\right)+\\tan \\left(\\frac{\\pi }{4}\\right)}{1-\\left(\\tan \\left(\\frac{\\pi }{6}\\right)\\right)\\left(\\tan \\left(\\frac{\\pi }{4}\\right)\\right)} \\end{align}[\/latex]<\/p>\r\nNext, we determine the individual tangents within the formula:\r\n<p style=\"text-align: center;\">[latex]\\tan \\left(\\frac{\\pi }{6}\\right)=\\frac{1}{\\sqrt{3}},\\tan \\left(\\frac{\\pi }{4}\\right)=1[\/latex]<\/p>\r\nSo we have\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\frac{\\pi }{6}+\\frac{\\pi }{4}\\right)&amp;=\\frac{\\frac{1}{\\sqrt{3}}+1}{1-\\left(\\frac{1}{\\sqrt{3}}\\right)\\left(1\\right)} \\\\ &amp;=\\frac{\\frac{1+\\sqrt{3}}{\\sqrt{3}}}{\\frac{\\sqrt{3}-1}{\\sqrt{3}}} \\\\ &amp;=\\frac{1+\\sqrt{3}}{\\sqrt{3}}\\left(\\frac{\\sqrt{3}}{\\sqrt{3}-1}\\right) \\\\ &amp;=\\frac{\\sqrt{3}+1}{\\sqrt{3}-1} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nFind the exact value of [latex]\\tan \\left(\\frac{2\\pi }{3}+\\frac{\\pi }{4}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"962695\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"962695\"]\r\n\r\n[latex]\\frac{1-\\sqrt{3}}{1+\\sqrt{3}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]173454[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 6: Finding Multiple Sums and Differences of Angles<\/h3>\r\nGiven [latex]\\text{ }\\sin \\alpha =\\frac{3}{5},0&lt;\\alpha &lt;\\frac{\\pi }{2},\\cos \\beta =-\\frac{5}{13},\\pi &lt;\\beta &lt;\\frac{3\\pi }{2}[\/latex], find\r\n<ol>\r\n \t<li>[latex]\\sin \\left(\\alpha +\\beta \\right)[\/latex]<\/li>\r\n \t<li>[latex]\\cos \\left(\\alpha +\\beta \\right)[\/latex]<\/li>\r\n \t<li>[latex]\\tan \\left(\\alpha +\\beta \\right)[\/latex]<\/li>\r\n \t<li>[latex]\\tan \\left(\\alpha -\\beta \\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"622511\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"622511\"]\r\n\r\nWe can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or tangent is provided for each of the individual angles. To do so, we construct what is called a reference triangle to help find each component of the sum and difference formulas.\r\n<ol>\r\n \t<li>To find [latex]\\sin \\left(\\alpha +\\beta \\right)[\/latex], we begin with [latex]\\sin \\alpha =\\frac{3}{5}[\/latex] and [latex]0&lt;\\alpha &lt;\\frac{\\pi }{2}[\/latex]. The side opposite [latex]\\alpha [\/latex] has length 3, the hypotenuse has length 5, and [latex]\\alpha [\/latex] is in the first quadrant. Using the Pythagorean Theorem, we can find the length of side [latex]a:[\/latex]\r\n<div style=\"text-align: center;\">\r\n\r\n[latex]\\begin{gathered}{a}^{2}+{3}^{2}={5}^{2} \\\\ {a}^{2}=16 \\\\ a=4 \\end{gathered}[\/latex]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164040\/CNX_Precalc_Figure_07_02_0032.jpg\" alt=\"Diagram of a triangle in the x,y plane. The vertices are at the origin, (4,0), and (4,3). The angle at the origin is alpha degrees, The angle formed by the x-axis and the side from (4,3) to (4,0) is a right angle. The side opposite the right angle has length 5.\" width=\"487\" height=\"252\" \/> <b>Figure 4<\/b>[\/caption]\r\n\r\n<\/div>\r\nSince [latex]\\cos \\beta =-\\frac{5}{13}[\/latex] and [latex]\\pi &lt;\\beta &lt;\\frac{3\\pi }{2}[\/latex], the side adjacent to [latex]\\beta [\/latex] is [latex]-5[\/latex], the hypotenuse is 13, and [latex]\\beta [\/latex] is in the third quadrant. Again, using the Pythagorean Theorem, we have<\/li>\r\n \t<li>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}{\\left(-5\\right)}^{2}+{a}^{2}&amp;={13}^{2} \\\\ 25+{a}^{2}&amp;=169 \\\\ {a}^{2}&amp;=144\\\\ a&amp;=\\pm 12 \\end{align}[\/latex]<\/p>\r\nSince [latex]\\beta [\/latex] is in the third quadrant, [latex]a=-12[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164043\/CNX_Precalc_Figure_07_02_0042.jpg\" alt=\"Diagram of a triangle in the x,y plane. The vertices are at the origin, (-5,0), and (-5, -12). The angle at the origin is Beta degrees. The angle formed by the x axis and the side from (-5, -12) to (-5,0) is a right angle. The side opposite the right angle has length 13.\" width=\"487\" height=\"568\" \/> <b>Figure 5<\/b>[\/caption]\r\n\r\nThe next step is finding the cosine of [latex]\\alpha [\/latex] and the sine of [latex]\\beta [\/latex]. The cosine of [latex]\\alpha [\/latex] is the adjacent side over the hypotenuse. We can find it from the triangle in Figure 5:\u00a0[latex]\\cos \\alpha =\\frac{4}{5}[\/latex]. We can also find the sine of [latex]\\beta [\/latex] from the triangle in Figure 5, as opposite side over the hypotenuse: [latex]\\sin \\beta =-\\frac{12}{13}[\/latex]. Now we are ready to evaluate [latex]\\sin \\left(\\alpha +\\beta \\right)[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left(\\alpha +\\beta \\right)&amp;=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta \\\\ &amp;=\\left(\\frac{3}{5}\\right)\\left(-\\frac{5}{13}\\right)+\\left(\\frac{4}{5}\\right)\\left(-\\frac{12}{13}\\right) \\\\ &amp;=-\\frac{15}{65}-\\frac{48}{65} \\\\ &amp;=-\\frac{63}{65} \\end{align}[\/latex]<\/div><\/li>\r\n \t<li>We can find [latex]\\cos \\left(\\alpha +\\beta \\right)[\/latex] in a similar manner. We substitute the values according to the formula.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha +\\beta \\right)&amp;=\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta \\\\ &amp;=\\left(\\frac{4}{5}\\right)\\left(-\\frac{5}{13}\\right)-\\left(\\frac{3}{5}\\right)\\left(-\\frac{12}{13}\\right) \\\\ &amp;=-\\frac{20}{65}+\\frac{36}{65} \\\\ &amp;=\\frac{16}{65} \\end{align}[\/latex]<\/div><\/li>\r\n \t<li>For [latex]\\tan \\left(\\alpha +\\beta \\right)[\/latex], if [latex]\\sin \\alpha =\\frac{3}{5}[\/latex] and [latex]\\cos \\alpha =\\frac{4}{5}[\/latex], then\r\n<div style=\"text-align: center;\">[latex]\\tan \\alpha =\\frac{\\frac{3}{5}}{\\frac{4}{5}}=\\frac{3}{4}[\/latex]<\/div>\r\nIf [latex]\\sin \\beta =-\\frac{12}{13}[\/latex] and [latex]\\cos \\beta =-\\frac{5}{13}[\/latex],\r\nthen\r\n<div style=\"text-align: center;\">[latex]\\tan \\beta =\\frac{\\frac{-12}{13}}{\\frac{-5}{13}}=\\frac{12}{5}[\/latex]<\/div>\r\nThen,\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\alpha +\\beta \\right)&amp;=\\frac{\\tan \\alpha +\\tan \\beta }{1-\\tan \\alpha \\tan \\beta } \\\\ &amp;=\\frac{\\frac{3}{4}+\\frac{12}{5}}{1-\\frac{3}{4}\\left(\\frac{12}{5}\\right)} \\\\ &amp;=\\frac{\\text{ }\\frac{63}{20}}{-\\frac{16}{20}} \\\\ &amp;=-\\frac{63}{16} \\end{align}[\/latex]<\/div><\/li>\r\n \t<li>To find [latex]\\tan \\left(\\alpha -\\beta \\right)[\/latex], we have the values we need. We can substitute them in and evaluate.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\alpha -\\beta \\right)&amp;=\\frac{\\tan \\alpha -\\tan \\beta }{1+\\tan \\alpha \\tan \\beta } \\\\ &amp;=\\frac{\\frac{3}{4}-\\frac{12}{5}}{1+\\frac{3}{4}\\left(\\frac{12}{5}\\right)} \\\\ &amp;=\\frac{-\\frac{33}{20}}{\\frac{56}{20}} \\\\ &amp;=-\\frac{33}{56}\\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n<h4>Analysis of the Solution<\/h4>\r\nA common mistake when addressing problems such as this one is that we may be tempted to think that [latex]\\alpha [\/latex] and [latex]\\beta [\/latex] are angles in the same triangle, which of course, they are not. Also note that\r\n<div style=\"text-align: center;\">[latex]\\tan \\left(\\alpha +\\beta \\right)=\\frac{\\sin \\left(\\alpha +\\beta \\right)}{\\cos \\left(\\alpha +\\beta \\right)}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>\r\n<div style=\"text-align: left;\"><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600; text-align: left;\">Use sum and difference formulas for cofunctions<\/span><\/div>\r\n<\/div>\r\nNow that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to do the same for their cofunctions. You may recall that if the sum of two positive angles is [latex]\\frac{\\pi }{2}[\/latex], those two angles are complements, and the sum of the two acute angles in a right triangle is [latex]\\frac{\\pi }{2}[\/latex], so they are also complements. In Figure 6, notice that if one of the acute angles is labeled as [latex]\\theta [\/latex], then the other acute angle must be labeled [latex]\\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164045\/CNX_Precalc_Figure_07_02_0072.jpg\" alt=\"Image of a right triangle. The remaining angles are labeled theta and pi\/2 - theta.\" width=\"487\" height=\"268\" \/> <b>Figure 6.<\/b> From these relationships, the cofunction identities are formed.[\/caption]\r\n\r\nNotice also that [latex]\\sin \\theta =\\cos \\left(\\frac{\\pi }{2}-\\theta \\right):[\/latex] opposite over hypotenuse. Thus, when two angles are complimentary, we can say that the sine of [latex]\\theta [\/latex] equals the <strong>cofunction<\/strong> of the complement of [latex]\\theta [\/latex]. Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.<span id=\"fs-id2872065\">\r\n<\/span>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Cofunction Identities<\/h3>\r\nThe cofunction identities are summarized in the table below.\r\n<table id=\"Table_07_02_02\" summary=\"Three rows, two columns\/ The table has ordered pairs of these row values: (sin(theta) = cos(pi\/2 - theta), cos(theta) = sin(pi\/2 - theta)), (tan(theta) = cot(pi\/2 - theta), cot(theta) = tan(pi\/2 - theta)), and (sec(theta) = csc(pi\/2 - theta), csc(theta) = sec(pi\/2 - theta)).\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\sin \\theta =\\cos \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\r\n<td>[latex]\\cos \\theta =\\sin \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\tan \\theta =\\cot \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\r\n<td>[latex]\\cot \\theta =\\tan \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\sec \\theta =\\csc \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\r\n<td>[latex]\\csc \\theta =\\sec \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNotice that the formulas in the table may also justified algebraically using the sum and difference formulas. For example, using\r\n<div style=\"text-align: center;\">[latex]\\cos \\left(\\alpha -\\beta \\right)=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta [\/latex],<\/div>\r\nwe can write\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\frac{\\pi }{2}-\\theta \\right)&amp;=\\cos \\frac{\\pi }{2}\\cos \\theta +\\sin \\frac{\\pi }{2}\\sin \\theta \\\\ &amp;=\\left(0\\right)\\cos \\theta +\\left(1\\right)\\sin \\theta \\\\ &amp;=\\sin \\theta\\end{align}[\/latex]<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 7: Finding a Cofunction with the Same Value as the Given Expression<\/h3>\r\nWrite [latex]\\tan \\frac{\\pi }{9}[\/latex] in terms of its cofunction.\r\n\r\n[reveal-answer q=\"406106\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"406106\"]\r\n\r\nThe cofunction of [latex]\\tan \\theta =\\cot \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]. Thus,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\frac{\\pi }{9}\\right)&amp;=\\cot \\left(\\frac{\\pi }{2}-\\frac{\\pi }{9}\\right) \\\\&amp; =\\cot \\left(\\frac{9\\pi }{18}-\\frac{2\\pi }{18}\\right) \\\\&amp; =\\cot \\left(\\frac{7\\pi }{18}\\right)\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nWrite [latex]\\sin \\frac{\\pi }{7}[\/latex] in terms of its cofunction.\r\n\r\n[reveal-answer q=\"615431\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"615431\"]\r\n\r\n[latex]\\cos \\left(\\frac{5\\pi }{14}\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]173455[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use sum and difference formulas to verify identities<\/h2>\r\nVerifying an identity means demonstrating that the equation holds for all values of the variable. It helps to be very familiar with the identities or to have a list of them accessible while working the problems.\r\n<div class=\"textbox\">\r\n<h3>How To: Given an identity, verify using sum and difference formulas.<\/h3>\r\n<ol>\r\n \t<li>Begin with the expression on the side of the equal sign that appears most complex. Rewrite that expression until it matches the other side of the equal sign. Occasionally, we might have to alter both sides, but working on only one side is the most efficient.<\/li>\r\n \t<li>Look for opportunities to use the sum and difference formulas.<\/li>\r\n \t<li>Rewrite sums or differences of quotients as single quotients.<\/li>\r\n \t<li>If the process becomes cumbersome, rewrite the expression in terms of sines and cosines.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div>\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 8: Verifying an Identity Involving Sine<\/h3>\r\nVerify the identity [latex]\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)=2\\sin \\alpha \\cos \\beta [\/latex].\r\n\r\n[reveal-answer q=\"464113\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"464113\"]\r\n\r\nWe see that the left side of the equation includes the sines of the sum and the difference of angles.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta\\\\ \\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta \\end{gathered}[\/latex]<\/p>\r\nWe can rewrite each using the sum and difference formulas.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)&amp;=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta +\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta \\\\ &amp;=2\\sin \\alpha \\cos \\beta \\end{align}[\/latex]<\/p>\r\nWe see that the identity is verified.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 9: Verifying an Identity Involving Tangent<\/h3>\r\nVerify the following identity.\r\n<p style=\"text-align: center;\">[latex]\\frac{\\sin \\left(\\alpha -\\beta \\right)}{\\cos \\alpha \\cos \\beta }=\\tan \\alpha -\\tan \\beta [\/latex]<\/p>\r\n[reveal-answer q=\"605610\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"605610\"]\r\n\r\nWe can begin by rewriting the numerator on the left side of the equation.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\sin \\left(\\alpha -\\beta \\right)}{\\cos \\alpha \\cos \\beta }&amp;=\\frac{\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta } \\\\ &amp;=\\frac{\\sin \\alpha \\cos \\beta}{\\cos \\alpha \\cos \\beta}-\\frac{\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta } &amp;&amp; \\text{Rewrite using a common denominator}. \\\\ &amp;=\\frac{\\sin \\alpha }{\\cos \\alpha }-\\frac{\\sin \\beta }{\\cos \\beta }&amp;&amp; \\text{Cancel}. \\\\ &amp;=\\tan \\alpha -\\tan \\beta &amp;&amp; \\text{Rewrite in terms of tangent}.\\end{align}[\/latex]<\/p>\r\nWe see that the identity is verified. In many cases, verifying tangent identities can successfully be accomplished by writing the tangent in terms of sine and cosine.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nVerify the identity: [latex]\\tan \\left(\\pi -\\theta \\right)=-\\tan \\theta [\/latex].\r\n\r\n[reveal-answer q=\"184004\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"184004\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\pi -\\theta \\right)&amp;=\\frac{\\tan \\left(\\pi \\right)-\\tan \\theta }{1+\\tan \\left(\\pi \\right)\\tan \\theta } \\\\ &amp;=\\frac{0-\\tan \\theta }{1+0\\cdot \\tan \\theta } \\\\ &amp;=-\\tan \\theta \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 10: Using Sum and Difference Formulas to Solve an Application Problem<\/h3>\r\nLet [latex]{L}_{1}[\/latex] and [latex]{L}_{2}[\/latex] denote two non-vertical intersecting lines, and let [latex]\\theta [\/latex] denote the acute angle between [latex]{L}_{1}[\/latex] and [latex]{L}_{2}[\/latex]. Show that\r\n<p style=\"text-align: center;\">[latex]\\tan \\theta =\\frac{{m}_{2}-{m}_{1}}{1+{m}_{1}{m}_{2}}[\/latex]<\/p>\r\nwhere [latex]{m}_{1}[\/latex] and [latex]{m}_{2}[\/latex] are the slopes of [latex]{L}_{1}[\/latex] and [latex]{L}_{2}[\/latex] respectively. (<strong>Hint:<\/strong> Use the fact that [latex]\\tan {\\theta }_{1}={m}_{1}[\/latex] and [latex]\\tan {\\theta }_{2}={m}_{2}[\/latex]. )\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164047\/CNX_Precalc_Figure_07_02_0052.jpg\" alt=\"Diagram of two non-vertical intersecting lines L1 and L2 also intersecting the x-axis. The acute angle formed by the intersection of L1 and L2 is theta. The acute angle formed by L2 and the x-axis is theta 1, and the acute angle formed by the x-axis and L1 is theta 2. \" width=\"487\" height=\"289\" \/> <b>Figure 7<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"429859\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"429859\"]\r\n\r\nUsing the difference formula for tangent, this problem does not seem as daunting as it might.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\theta &amp;=\\tan \\left({\\theta }_{2}-{\\theta }_{1}\\right) \\\\ &amp;=\\frac{\\tan {\\theta }_{2}-\\tan {\\theta }_{1}}{1+\\tan {\\theta }_{1}\\tan {\\theta }_{2}} \\\\ &amp;=\\frac{{m}_{2}-{m}_{1}}{1+{m}_{1}{m}_{2}}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]<span style=\"text-align: center; font-size: 0.9em;\">\u00a0<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 11: Investigating a Guy-wire Problem<\/h3>\r\n[caption id=\"\" align=\"aligncenter\" width=\"590\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164049\/CNX_Precalc_Figure_07_02_0062.jpg\" alt=\"Two right triangles. Both share the same base, 50 feet. The first has a height of 40 ft and hypotenuse S. The second has height 47 ft and hypotenuse R. The height sides of the triangles are overlapping. There is a B degree angle between R and the base, and an a degree angle between the two hypotenuses within the B degree angle. \" width=\"590\" height=\"322\" \/> <b>Figure 8<\/b>[\/caption]\r\n\r\nFor a climbing wall, a guy-wire [latex]R[\/latex] is attached 47 feet high on a vertical pole. Added support is provided by another guy-wire [latex]S[\/latex] attached 40 feet above ground on the same pole. If the wires are attached to the ground 50 feet from the pole, find the angle [latex]\\alpha [\/latex] between the wires.\u00a0<span id=\"fs-id1386435\">\r\n<\/span>\r\n\r\n[reveal-answer q=\"48078\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"48078\"]\r\n\r\nLet\u2019s first summarize the information we can gather from the diagram. As only the sides adjacent to the right angle are known, we can use the tangent function. Notice that [latex]\\tan \\beta =\\frac{47}{50}[\/latex], and [latex]\\tan \\left(\\beta -\\alpha \\right)=\\frac{40}{50}=\\frac{4}{5}[\/latex]. We can then use difference formula for tangent.\r\n<p style=\"text-align: center;\">[latex]\\tan \\left(\\beta -\\alpha \\right)=\\frac{\\tan \\beta -\\tan \\alpha }{1+\\tan \\beta \\tan \\alpha }[\/latex]<\/p>\r\nNow, substituting the values we know into the formula, we have\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\frac{4}{5}=\\frac{\\frac{47}{50}-\\tan \\alpha }{1+\\frac{47}{50}\\tan \\alpha } \\\\ 4\\left(1+\\frac{47}{50}\\tan \\alpha \\right)=5\\left(\\frac{47}{50}-\\tan \\alpha \\right) \\end{gathered}[\/latex]<\/p>\r\nUse the distributive property, and then simplify the functions.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}4\\left(1\\right)+4\\left(\\frac{47}{50}\\right)\\tan \\alpha =5\\left(\\frac{47}{50}\\right)-5\\tan \\alpha\\\\ 4+3.76\\tan \\alpha =4.7 - 5\\tan \\alpha \\\\ 5\\tan \\alpha +3.76\\tan \\alpha =0.7\\\\ 8.76\\tan \\alpha =0.7\\\\ \\tan \\alpha \\approx 0.07991 \\\\ {\\tan }^{-1}\\left(0.07991\\right)\\approx .079741 \\end{gathered}[\/latex]<\/p>\r\nNow we can calculate the angle in degrees.\r\n<p style=\"text-align: center;\">[latex]\\alpha \\approx 0.079741\\left(\\frac{180}{\\pi }\\right)\\approx {4.57}^{\\circ }[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nOccasionally, when an application appears that includes a right triangle, we may think that solving is a matter of applying the Pythagorean Theorem. That may be partially true, but it depends on what the problem is asking and what information is given.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>\r\n\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Key Equations<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Sum Formula for Cosine<\/td>\r\n<td>[latex]\\cos \\left(\\alpha +\\beta \\right)=\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Difference Formula for Cosine<\/td>\r\n<td>[latex]\\cos \\left(\\alpha -\\beta \\right)=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Sum Formula for Sine<\/td>\r\n<td>[latex]\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Difference Formula for Sine<\/td>\r\n<td>[latex]\\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Sum Formula for Tangent<\/td>\r\n<td>[latex]\\tan \\left(\\alpha +\\beta \\right)=\\frac{\\tan \\alpha +\\tan \\beta }{1-\\tan \\alpha \\tan \\beta }[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Difference Formula for Tangent<\/td>\r\n<td>[latex]\\tan \\left(\\alpha -\\beta \\right)=\\frac{\\tan \\alpha -\\tan \\beta }{1+\\tan \\alpha \\tan \\beta }[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cofunction identities<\/td>\r\n<td>[latex]\\sin \\theta =\\cos \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]\r\n\r\n[latex]\\cos \\theta =\\sin \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]\r\n\r\n[latex]\\tan \\theta =\\cot \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]\r\n\r\n[latex]\\cot \\theta =\\tan \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]\r\n\r\n[latex]\\sec \\theta =\\csc \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]\r\n\r\n[latex]\\csc \\theta =\\sec \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>The sum formula for cosines states that the cosine of the sum of two angles equals the product of the cosines of the angles minus the product of the sines of the angles. The difference formula for cosines states that the cosine of the difference of two angles equals the product of the cosines of the angles plus the product of the sines of the angles.<\/li>\r\n \t<li>The sum and difference formulas can be used to find the exact values of the sine, cosine, or tangent of an angle.<\/li>\r\n \t<li>The sum formula for sines states that the sine of the sum of two angles equals the product of the sine of the first angle and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second angle. The difference formula for sines states that the sine of the difference of two angles equals the product of the sine of the first angle and cosine of the second angle minus the product of the cosine of the first angle and the sine of the second angle.<\/li>\r\n \t<li>The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions.<\/li>\r\n \t<li>The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by 1 minus the product of the tangents of the angles. The difference formula for tangent states that the tangent of the difference of two angles equals the difference of the tangents of the angles divided by 1 plus the product of the tangents of the angles.<\/li>\r\n \t<li>The Pythagorean Theorem along with the sum and difference formulas can be used to find multiple sums and differences of angles.<\/li>\r\n \t<li>The cofunction identities apply to complementary angles and pairs of reciprocal functions.<\/li>\r\n \t<li>Sum and difference formulas are useful in verifying identities.<\/li>\r\n \t<li>Application problems are often easier to solve by using sum and difference formulas.<\/li>\r\n<\/ul>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li style=\"font-weight: 400;\">Use sum and difference formulas for cosine.<\/li>\n<li style=\"font-weight: 400;\">Use sum and difference formulas for sine.<\/li>\n<li style=\"font-weight: 400;\">Use sum and difference formulas for tangent.<\/li>\n<li style=\"font-weight: 400;\">Use sum and difference formulas for cofunctions.<\/li>\n<li style=\"font-weight: 400;\">Use sum and difference formulas to verify identities.<\/li>\n<\/ul>\n<\/div>\n<h2>Use sum and difference formulas for cosine<\/h2>\n<p>Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values. We can use the <strong>special angles<\/strong>, which we can review in the unit circle shown in Figure 2.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164036\/CNX_Precalc_Figure_07_01_0042.jpg\" alt=\"Diagram of the unit circle with points labeled on its edge. P point is at an angle a from the positive x axis with coordinates (cosa, sina). Point Q is at an angle of B from the positive x axis with coordinates (cosb, sinb). Angle POQ is a - B degrees. Point A is at an angle of (a-B) from the x axis with coordinates (cos(a-B), sin(a-B)). Point B is just at point (1,0). Angle AOB is also a - B degrees. Radii PO, AO, QO, and BO are all 1 unit long and are the legs of triangles POQ and AOB. Triangle POQ is a rotation of triangle AOB, so the distance from P to Q is the same as the distance from A to B.\" width=\"975\" height=\"638\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2.<\/b> The Unit Circle<\/p>\n<\/div>\n<p>We will begin with the <strong>sum and difference formulas for cosine<\/strong>, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles.<\/p>\n<table id=\"Table_07_02_01\" summary=\"Two rows, two columns. The table has ordered pairs of these row values: (Sum formula for cosine, cos(a+B) = cos(a)cos(B) - sin(a)sin(B)) and (Difference formula for cosine, cos(a-B) = cos(a)cos(B) + sin(a)sin(B)).\">\n<tbody>\n<tr>\n<td><strong>Sum formula for cosine<\/strong><\/td>\n<td>[latex]\\cos \\left(\\alpha +\\beta \\right)=\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Difference formula for cosine<\/strong><\/td>\n<td>[latex]\\cos \\left(\\alpha -\\beta \\right)=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>First, we will prove the difference formula for cosines. Let\u2019s consider two points on the unit circle. Point [latex]P[\/latex] is at an angle [latex]\\alpha[\/latex] from the positive <em>x-<\/em>axis with coordinates [latex]\\left(\\cos \\alpha ,\\sin \\alpha \\right)[\/latex] and point [latex]Q[\/latex] is at an angle of [latex]\\beta[\/latex] from the positive <em>x-<\/em>axis with coordinates [latex]\\left(\\cos \\beta ,\\sin \\beta \\right)[\/latex]. Note the measure of angle [latex]POQ[\/latex] is [latex]\\alpha -\\beta[\/latex].<\/p>\n<p>Label two more points: [latex]A[\/latex] at an angle of [latex]\\left(\\alpha -\\beta \\right)[\/latex] from the positive <em>x-<\/em>axis with coordinates [latex]\\left(\\cos \\left(\\alpha -\\beta \\right),\\sin \\left(\\alpha -\\beta \\right)\\right)[\/latex]; and point [latex]B[\/latex] with coordinates [latex]\\left(1,0\\right)[\/latex]. Triangle [latex]POQ[\/latex] is a rotation of triangle [latex]AOB[\/latex] and thus the distance from [latex]P[\/latex] to [latex]Q[\/latex] is the same as the distance from [latex]A[\/latex] to [latex]B[\/latex].<\/p>\n<figure id=\"Figure_07_02_002\" class=\"small\"><span id=\"fs-id1130636\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164038\/CNX_Precalc_Figure_07_02_0022.jpg\" alt=\"Diagram of the unit circle with points labeled on its edge. P point is at an angle a from the positive x axis with coordinates (cosa, sina). Point Q is at an angle of B from the positive x axis with coordinates (cosb, sinb). Angle POQ is a - B degrees. Point A is at an angle of (a-B) from the x axis with coordinates (cos(a-B), sin(a-B)). Point B is just at point (1,0). Angle AOB is also a - B degrees. Radii PO, AO, QO, and BO are all 1 unit long and are the legs of triangles POQ and AOB. Triangle POQ is a rotation of triangle AOB, so the distance from P to Q is the same as the distance from A to B.\" \/><\/span><\/figure>\n<p style=\"text-align: center;\"><strong>Figure 3.\u00a0<\/strong>We can find the distance from [latex]P[\/latex] to [latex]Q[\/latex] using the <strong>distance formula<\/strong>.<\/p>\n<div><\/div>\n<div style=\"text-align: center;\">[latex]\\begin{align}{d}_{PQ}&=\\sqrt{{\\left(\\cos \\alpha -\\cos \\beta \\right)}^{2}+{\\left(\\sin \\alpha -\\sin \\beta \\right)}^{2}} \\\\ &=\\sqrt{{\\cos }^{2}\\alpha -2\\cos \\alpha \\cos \\beta +{\\cos }^{2}\\beta +{\\sin }^{2}\\alpha -2\\sin \\alpha \\sin \\beta +{\\sin }^{2}\\beta } \\end{align}[\/latex]<\/div>\n<p>Then we apply the <strong>Pythagorean identity<\/strong> and simplify.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align} &=\\sqrt{\\left({\\cos }^{2}\\alpha +{\\sin }^{2}\\alpha \\right)+\\left({\\cos }^{2}\\beta +{\\sin }^{2}\\beta \\right)-2\\cos \\alpha \\cos \\beta -2\\sin \\alpha \\sin \\beta } \\\\ &=\\sqrt{1+1 - 2\\cos \\alpha \\cos \\beta -2\\sin \\alpha \\sin \\beta } \\\\ &=\\sqrt{2 - 2\\cos \\alpha \\cos \\beta -2\\sin \\alpha \\sin \\beta } \\end{align}[\/latex]<\/div>\n<p>Similarly, using the distance formula we can find the distance from [latex]A[\/latex] to [latex]B[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}{d}_{AB}&=\\sqrt{{\\left(\\cos \\left(\\alpha -\\beta \\right)-1\\right)}^{2}+{\\left(\\sin \\left(\\alpha -\\beta \\right)-0\\right)}^{2}} \\\\ &=\\sqrt{{\\cos }^{2}\\left(\\alpha -\\beta \\right)-2\\cos \\left(\\alpha -\\beta \\right)+1+{\\sin }^{2}\\left(\\alpha -\\beta \\right)} \\end{align}[\/latex]<\/div>\n<p>Applying the Pythagorean identity and simplifying we get:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align} &=\\sqrt{\\left({\\cos }^{2}\\left(\\alpha -\\beta \\right)+{\\sin }^{2}\\left(\\alpha -\\beta \\right)\\right)-2\\cos \\left(\\alpha -\\beta \\right)+1}\\\\ &=\\sqrt{1 - 2\\cos \\left(\\alpha -\\beta \\right)+1}\\\\ &=\\sqrt{2 - 2\\cos \\left(\\alpha -\\beta \\right)} \\end{align}[\/latex]<\/div>\n<p>Because the two distances are the same, we set them equal to each other and simplify.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align} \\sqrt{2 - 2\\cos \\alpha \\cos \\beta -2\\sin \\alpha \\sin \\beta }&=\\sqrt{2 - 2\\cos \\left(\\alpha -\\beta \\right)} \\\\ 2 - 2\\cos \\alpha \\cos \\beta -2\\sin \\alpha \\sin \\beta &=2 - 2\\cos \\left(\\alpha -\\beta \\right) \\end{align}[\/latex]<\/div>\n<p>Finally we subtract [latex]2[\/latex] from both sides and divide both sides by [latex]-2[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta =\\cos \\left(\\alpha -\\beta \\right)\\end{align}[\/latex]<\/div>\n<p>Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Sum and Difference Formulas for Cosine<\/h3>\n<p>These formulas can be used to calculate the cosine of sums and differences of angles.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha +\\beta \\right)=\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta\\end{align}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha -\\beta \\right)=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta\\end{align}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given two angles, find the cosine of the difference between the angles.<strong><br \/>\n<\/strong><\/h3>\n<ol id=\"fs-id1374215\">\n<li>Write the difference formula for cosine.<\/li>\n<li>Substitute the values of the given angles into the formula.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Finding the Exact Value Using the Formula for the Cosine of the Difference of Two Angles<\/h3>\n<p>Using the formula for the cosine of the difference of two angles, find the exact value of [latex]\\cos \\left(\\frac{5\\pi }{4}-\\frac{\\pi }{6}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q956915\">Show Solution<\/span><\/p>\n<div id=\"q956915\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the formula for the cosine of the difference of two angles. We have<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha -\\beta \\right)&=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta \\\\ \\cos \\left(\\frac{5\\pi }{4}-\\frac{\\pi }{6}\\right)&=\\cos \\left(\\frac{5\\pi }{4}\\right)\\cos \\left(\\frac{\\pi }{6}\\right)+\\sin \\left(\\frac{5\\pi }{4}\\right)\\sin \\left(\\frac{\\pi }{6}\\right) \\\\ &=\\left(-\\frac{\\sqrt{2}}{2}\\right)\\left(\\frac{\\sqrt{3}}{2}\\right)-\\left(\\frac{\\sqrt{2}}{2}\\right)\\left(\\frac{1}{2}\\right) \\\\ &=-\\frac{\\sqrt{6}}{4}-\\frac{\\sqrt{2}}{4} \\\\ &=\\frac{-\\sqrt{6}-\\sqrt{2}}{4} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the exact value of [latex]\\cos \\left(\\frac{\\pi }{3}-\\frac{\\pi }{4}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q700317\">Show Solution<\/span><\/p>\n<div id=\"q700317\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{\\sqrt{2}+\\sqrt{6}}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Finding the Exact Value Using the Formula for the Sum of Two Angles for Cosine<\/h3>\n<p>Find the exact value of [latex]\\cos \\left({75}^{\\circ }\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q617422\">Show Solution<\/span><\/p>\n<div id=\"q617422\" class=\"hidden-answer\" style=\"display: none\">\n<p>As [latex]{75}^{\\circ }={45}^{\\circ }+{30}^{\\circ }[\/latex], we can evaluate [latex]\\cos \\left({75}^{\\circ }\\right)[\/latex] as [latex]\\cos \\left({45}^{\\circ }+{30}^{\\circ }\\right)[\/latex]. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left({45}^{\\circ }+{30}^{\\circ }\\right)&=\\cos \\left({45}^{\\circ }\\right)\\cos \\left({30}^{\\circ }\\right)-\\sin \\left({45}^{\\circ }\\right)\\sin \\left({30}^{\\circ }\\right) \\\\ &=\\frac{\\sqrt{2}}{2}\\left(\\frac{\\sqrt{3}}{2}\\right)-\\frac{\\sqrt{2}}{2}\\left(\\frac{1}{2}\\right) \\\\ &=\\frac{\\sqrt{6}}{4}-\\frac{\\sqrt{2}}{4} \\\\ &=\\frac{\\sqrt{6}-\\sqrt{2}}{4} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the exact value of [latex]\\cos \\left({105}^{\\circ }\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q683323\">Show Solution<\/span><\/p>\n<div id=\"q683323\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{\\sqrt{2}-\\sqrt{6}}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm173445\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173445&theme=oea&iframe_resize_id=ohm173445\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use sum and difference formulas for sine<\/h2>\n<p>The <strong>sum and difference formulas for sine<\/strong> can be derived in the same manner as those for cosine, and they resemble the cosine formulas.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Sum and Difference Formulas for Sine<\/h3>\n<p>These formulas can be used to calculate the sines of sums and differences of angles.<\/p>\n<p style=\"text-align: center;\">[latex]\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given two angles, find the sine of the difference between the angles.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Write the difference formula for sine.<\/li>\n<li>Substitute the given angles into the formula.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Using Sum and Difference Identities to Evaluate the Difference of Angles<\/h3>\n<p>Use the sum and difference identities to evaluate the difference of the angles and show that part <em>a<\/em> equals part <em>b.<\/em><\/p>\n<ol>\n<li>[latex]\\sin \\left({45}^{\\circ }-{30}^{\\circ }\\right)[\/latex]<\/li>\n<li>[latex]\\sin \\left({135}^{\\circ }-{120}^{\\circ }\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q191680\">Show Solution<\/span><\/p>\n<div id=\"q191680\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Let\u2019s begin by writing the formula and substitute the given angles.\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left(\\alpha -\\beta \\right)&=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta \\\\ \\sin \\left({45}^{\\circ }-{30}^{\\circ }\\right)&=\\sin \\left({45}^{\\circ }\\right)\\cos \\left({30}^{\\circ }\\right)-\\cos \\left({45}^{\\circ }\\right)\\sin \\left({30}^{\\circ }\\right) \\end{align}[\/latex]<\/div>\n<p>Next, we need to find the values of the trigonometric expressions.<\/p>\n<div style=\"text-align: center;\">[latex]\\sin \\left({45}^{\\circ }\\right)=\\frac{\\sqrt{2}}{2},\\text{ }\\cos \\left({30}^{\\circ }\\right)=\\frac{\\sqrt{3}}{2},\\text{ }\\cos \\left({45}^{\\circ }\\right)=\\frac{\\sqrt{2}}{2},\\text{ }\\sin \\left({30}^{\\circ }\\right)=\\frac{1}{2}[\/latex]<\/div>\n<p>Now we can substitute these values into the equation and simplify.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align} \\sin \\left({45}^{\\circ }-{30}^{\\circ }\\right)&=\\frac{\\sqrt{2}}{2}\\left(\\frac{\\sqrt{3}}{2}\\right)-\\frac{\\sqrt{2}}{2}\\left(\\frac{1}{2}\\right) \\\\ &=\\frac{\\sqrt{6}-\\sqrt{2}}{4}\\hfill \\end{align}[\/latex]<\/div>\n<\/li>\n<li>Again, we write the formula and substitute the given angles.\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left(\\alpha -\\beta \\right)&=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta\\\\ \\sin \\left({135}^{\\circ }-{120}^{\\circ }\\right)&=\\sin \\left({135}^{\\circ }\\right)\\cos \\left({120}^{\\circ }\\right)-\\cos \\left({135}^{\\circ }\\right)\\sin \\left({120}^{\\circ }\\right)\\end{align}[\/latex]<\/div>\n<p>Next, we find the values of the trigonometric expressions.<\/p>\n<div style=\"text-align: center;\">[latex]\\sin \\left({135}^{\\circ }\\right)=\\frac{\\sqrt{2}}{2},\\cos \\left({120}^{\\circ }\\right)=-\\frac{1}{2},\\cos \\left({135}^{\\circ }\\right)=\\frac{\\sqrt{2}}{2},\\sin \\left({120}^{\\circ }\\right)=\\frac{\\sqrt{3}}{2}[\/latex]<\/div>\n<p>Now we can substitute these values into the equation and simplify.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left({135}^{\\circ }-{120}^{\\circ }\\right)&=\\frac{\\sqrt{2}}{2}\\left(-\\frac{1}{2}\\right)-\\left(-\\frac{\\sqrt{2}}{2}\\right)\\left(\\frac{\\sqrt{3}}{2}\\right) \\\\ &=\\frac{-\\sqrt{2}+\\sqrt{6}}{4} \\\\ &=\\frac{\\sqrt{6}-\\sqrt{2}}{4} \\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Finding the Exact Value of an Expression Involving an Inverse Trigonometric Function<\/h3>\n<p>Find the exact value of [latex]\\sin \\left({\\cos }^{-1}\\frac{1}{2}+{\\sin }^{-1}\\frac{3}{5}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q954448\">Show Solution<\/span><\/p>\n<div id=\"q954448\" class=\"hidden-answer\" style=\"display: none\">\n<p>The pattern displayed in this problem is [latex]\\sin \\left(\\alpha +\\beta \\right)[\/latex]. Let [latex]\\alpha ={\\cos }^{-1}\\frac{1}{2}[\/latex] and [latex]\\beta ={\\sin }^{-1}\\frac{3}{5}[\/latex]. Then we can write<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} \\cos \\alpha &=\\frac{1}{2},0\\le \\alpha \\le \\pi\\\\ \\sin \\beta &=\\frac{3}{5},-\\frac{\\pi }{2}\\le \\beta \\le \\frac{\\pi }{2} \\end{align}[\/latex]<\/p>\n<p>We will use the Pythagorean identities to find [latex]\\sin \\alpha[\/latex] and [latex]\\cos \\beta[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sin \\alpha &=\\sqrt{1-{\\cos }^{2}\\alpha } \\\\ &=\\sqrt{1-\\frac{1}{4}} \\\\ &=\\sqrt{\\frac{3}{4}} \\\\ &=\\frac{\\sqrt{3}}{2} \\\\ \\cos \\beta &=\\sqrt{1-{\\sin }^{2}\\beta } \\\\ &=\\sqrt{1-\\frac{9}{25}} \\\\ &=\\sqrt{\\frac{16}{25}} \\\\ &=\\frac{4}{5}\\end{align}[\/latex]<\/p>\n<p>Using the sum formula for sine,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left({\\cos }^{-1}\\frac{1}{2}+{\\sin }^{-1}\\frac{3}{5}\\right)&=\\sin \\left(\\alpha +\\beta \\right) \\\\ &=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta \\\\ &=\\frac{\\sqrt{3}}{2}\\cdot \\frac{4}{5}+\\frac{1}{2}\\cdot \\frac{3}{5} \\\\ &=\\frac{4\\sqrt{3}+3}{10}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm173443\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173443&theme=oea&iframe_resize_id=ohm173443\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use sum and difference formulas for tangent<\/h2>\n<p>Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.<\/p>\n<p>Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall, [latex]\\tan x=\\frac{\\sin x}{\\cos x},\\cos x\\ne 0[\/latex].<\/p>\n<p>Let\u2019s derive the sum formula for tangent.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\alpha +\\beta \\right)&=\\frac{\\sin \\left(\\alpha +\\beta \\right)}{\\cos \\left(\\alpha +\\beta \\right)} \\\\[1mm] &=\\frac{\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta } \\\\[1mm] &=\\frac{\\frac{\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }}{\\frac{\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }} && \\text{Divide the numerator and denominator by cos}\\alpha \\text{cos}\\beta \\\\[1mm] &=\\frac{\\frac{\\sin \\alpha \\cos \\beta }{\\cos \\alpha \\cos \\beta}+\\frac{\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }}{\\frac{\\cos \\alpha \\cos \\beta }{\\cos \\alpha \\cos \\beta }-\\frac{\\sin \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }} && \\text{Split the fractions.} \\\\[1mm] &=\\frac{\\frac{\\sin \\alpha }{\\cos \\alpha }+\\frac{\\sin \\beta }{\\cos \\beta }}{1-\\frac{\\sin \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }} && \\text{Cancel.} \\\\[1mm] &=\\frac{\\tan \\alpha +\\tan \\beta }{1-\\tan \\alpha \\tan \\beta } \\end{align}[\/latex]<\/div>\n<p>We can derive the difference formula for tangent in a similar way.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Sum and Difference Formulas for Tangent<\/h3>\n<p>The <strong>sum and difference formulas for tangent<\/strong> are:<\/p>\n<p style=\"text-align: center;\">[latex]\\tan \\left(\\alpha +\\beta \\right)=\\frac{\\tan \\alpha +\\tan \\beta }{1-\\tan \\alpha \\tan \\beta }[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\tan \\left(\\alpha -\\beta \\right)=\\frac{\\tan \\alpha -\\tan \\beta }{1+\\tan \\alpha \\tan \\beta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given two angles, find the tangent of the sum of the angles.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Write the sum formula for tangent.<\/li>\n<li>Substitute the given angles into the formula.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 5: Finding the Exact Value of an Expression Involving Tangent<\/h3>\n<p>Find the exact value of [latex]\\tan \\left(\\frac{\\pi }{6}+\\frac{\\pi }{4}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q796417\">Show Solution<\/span><\/p>\n<div id=\"q796417\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let\u2019s first write the sum formula for tangent and substitute the given angles into the formula.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\alpha +\\beta \\right)&=\\frac{\\tan \\alpha +\\tan \\beta }{1-\\tan \\alpha \\tan \\beta }\\\\ \\tan \\left(\\frac{\\pi }{6}+\\frac{\\pi }{4}\\right)&=\\frac{\\tan \\left(\\frac{\\pi }{6}\\right)+\\tan \\left(\\frac{\\pi }{4}\\right)}{1-\\left(\\tan \\left(\\frac{\\pi }{6}\\right)\\right)\\left(\\tan \\left(\\frac{\\pi }{4}\\right)\\right)} \\end{align}[\/latex]<\/p>\n<p>Next, we determine the individual tangents within the formula:<\/p>\n<p style=\"text-align: center;\">[latex]\\tan \\left(\\frac{\\pi }{6}\\right)=\\frac{1}{\\sqrt{3}},\\tan \\left(\\frac{\\pi }{4}\\right)=1[\/latex]<\/p>\n<p>So we have<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\frac{\\pi }{6}+\\frac{\\pi }{4}\\right)&=\\frac{\\frac{1}{\\sqrt{3}}+1}{1-\\left(\\frac{1}{\\sqrt{3}}\\right)\\left(1\\right)} \\\\ &=\\frac{\\frac{1+\\sqrt{3}}{\\sqrt{3}}}{\\frac{\\sqrt{3}-1}{\\sqrt{3}}} \\\\ &=\\frac{1+\\sqrt{3}}{\\sqrt{3}}\\left(\\frac{\\sqrt{3}}{\\sqrt{3}-1}\\right) \\\\ &=\\frac{\\sqrt{3}+1}{\\sqrt{3}-1} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the exact value of [latex]\\tan \\left(\\frac{2\\pi }{3}+\\frac{\\pi }{4}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q962695\">Show Solution<\/span><\/p>\n<div id=\"q962695\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{1-\\sqrt{3}}{1+\\sqrt{3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm173454\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173454&theme=oea&iframe_resize_id=ohm173454\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 6: Finding Multiple Sums and Differences of Angles<\/h3>\n<p>Given [latex]\\text{ }\\sin \\alpha =\\frac{3}{5},0<\\alpha <\\frac{\\pi }{2},\\cos \\beta =-\\frac{5}{13},\\pi <\\beta <\\frac{3\\pi }{2}[\/latex], find\n\n\n<ol>\n<li>[latex]\\sin \\left(\\alpha +\\beta \\right)[\/latex]<\/li>\n<li>[latex]\\cos \\left(\\alpha +\\beta \\right)[\/latex]<\/li>\n<li>[latex]\\tan \\left(\\alpha +\\beta \\right)[\/latex]<\/li>\n<li>[latex]\\tan \\left(\\alpha -\\beta \\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q622511\">Show Solution<\/span><\/p>\n<div id=\"q622511\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or tangent is provided for each of the individual angles. To do so, we construct what is called a reference triangle to help find each component of the sum and difference formulas.<\/p>\n<ol>\n<li>To find [latex]\\sin \\left(\\alpha +\\beta \\right)[\/latex], we begin with [latex]\\sin \\alpha =\\frac{3}{5}[\/latex] and [latex]0<\\alpha <\\frac{\\pi }{2}[\/latex]. The side opposite [latex]\\alpha[\/latex] has length 3, the hypotenuse has length 5, and [latex]\\alpha[\/latex] is in the first quadrant. Using the Pythagorean Theorem, we can find the length of side [latex]a:[\/latex]\n\n\n<div style=\"text-align: center;\">\n<p>[latex]\\begin{gathered}{a}^{2}+{3}^{2}={5}^{2} \\\\ {a}^{2}=16 \\\\ a=4 \\end{gathered}[\/latex]<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164040\/CNX_Precalc_Figure_07_02_0032.jpg\" alt=\"Diagram of a triangle in the x,y plane. The vertices are at the origin, (4,0), and (4,3). The angle at the origin is alpha degrees, The angle formed by the x-axis and the side from (4,3) to (4,0) is a right angle. The side opposite the right angle has length 5.\" width=\"487\" height=\"252\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<\/div>\n<p>Since [latex]\\cos \\beta =-\\frac{5}{13}[\/latex] and [latex]\\pi <\\beta <\\frac{3\\pi }{2}[\/latex], the side adjacent to [latex]\\beta[\/latex] is [latex]-5[\/latex], the hypotenuse is 13, and [latex]\\beta[\/latex] is in the third quadrant. Again, using the Pythagorean Theorem, we have<\/li>\n<li>\n<p style=\"text-align: center;\">[latex]\\begin{align}{\\left(-5\\right)}^{2}+{a}^{2}&={13}^{2} \\\\ 25+{a}^{2}&=169 \\\\ {a}^{2}&=144\\\\ a&=\\pm 12 \\end{align}[\/latex]<\/p>\n<p>Since [latex]\\beta[\/latex] is in the third quadrant, [latex]a=-12[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164043\/CNX_Precalc_Figure_07_02_0042.jpg\" alt=\"Diagram of a triangle in the x,y plane. The vertices are at the origin, (-5,0), and (-5, -12). The angle at the origin is Beta degrees. The angle formed by the x axis and the side from (-5, -12) to (-5,0) is a right angle. The side opposite the right angle has length 13.\" width=\"487\" height=\"568\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<p>The next step is finding the cosine of [latex]\\alpha[\/latex] and the sine of [latex]\\beta[\/latex]. The cosine of [latex]\\alpha[\/latex] is the adjacent side over the hypotenuse. We can find it from the triangle in Figure 5:\u00a0[latex]\\cos \\alpha =\\frac{4}{5}[\/latex]. We can also find the sine of [latex]\\beta[\/latex] from the triangle in Figure 5, as opposite side over the hypotenuse: [latex]\\sin \\beta =-\\frac{12}{13}[\/latex]. Now we are ready to evaluate [latex]\\sin \\left(\\alpha +\\beta \\right)[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left(\\alpha +\\beta \\right)&=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta \\\\ &=\\left(\\frac{3}{5}\\right)\\left(-\\frac{5}{13}\\right)+\\left(\\frac{4}{5}\\right)\\left(-\\frac{12}{13}\\right) \\\\ &=-\\frac{15}{65}-\\frac{48}{65} \\\\ &=-\\frac{63}{65} \\end{align}[\/latex]<\/div>\n<\/li>\n<li>We can find [latex]\\cos \\left(\\alpha +\\beta \\right)[\/latex] in a similar manner. We substitute the values according to the formula.\n<div style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha +\\beta \\right)&=\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta \\\\ &=\\left(\\frac{4}{5}\\right)\\left(-\\frac{5}{13}\\right)-\\left(\\frac{3}{5}\\right)\\left(-\\frac{12}{13}\\right) \\\\ &=-\\frac{20}{65}+\\frac{36}{65} \\\\ &=\\frac{16}{65} \\end{align}[\/latex]<\/div>\n<\/li>\n<li>For [latex]\\tan \\left(\\alpha +\\beta \\right)[\/latex], if [latex]\\sin \\alpha =\\frac{3}{5}[\/latex] and [latex]\\cos \\alpha =\\frac{4}{5}[\/latex], then\n<div style=\"text-align: center;\">[latex]\\tan \\alpha =\\frac{\\frac{3}{5}}{\\frac{4}{5}}=\\frac{3}{4}[\/latex]<\/div>\n<p>If [latex]\\sin \\beta =-\\frac{12}{13}[\/latex] and [latex]\\cos \\beta =-\\frac{5}{13}[\/latex],<br \/>\nthen<\/p>\n<div style=\"text-align: center;\">[latex]\\tan \\beta =\\frac{\\frac{-12}{13}}{\\frac{-5}{13}}=\\frac{12}{5}[\/latex]<\/div>\n<p>Then,<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\alpha +\\beta \\right)&=\\frac{\\tan \\alpha +\\tan \\beta }{1-\\tan \\alpha \\tan \\beta } \\\\ &=\\frac{\\frac{3}{4}+\\frac{12}{5}}{1-\\frac{3}{4}\\left(\\frac{12}{5}\\right)} \\\\ &=\\frac{\\text{ }\\frac{63}{20}}{-\\frac{16}{20}} \\\\ &=-\\frac{63}{16} \\end{align}[\/latex]<\/div>\n<\/li>\n<li>To find [latex]\\tan \\left(\\alpha -\\beta \\right)[\/latex], we have the values we need. We can substitute them in and evaluate.\n<div style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\alpha -\\beta \\right)&=\\frac{\\tan \\alpha -\\tan \\beta }{1+\\tan \\alpha \\tan \\beta } \\\\ &=\\frac{\\frac{3}{4}-\\frac{12}{5}}{1+\\frac{3}{4}\\left(\\frac{12}{5}\\right)} \\\\ &=\\frac{-\\frac{33}{20}}{\\frac{56}{20}} \\\\ &=-\\frac{33}{56}\\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<h4>Analysis of the Solution<\/h4>\n<p>A common mistake when addressing problems such as this one is that we may be tempted to think that [latex]\\alpha[\/latex] and [latex]\\beta[\/latex] are angles in the same triangle, which of course, they are not. Also note that<\/p>\n<div style=\"text-align: center;\">[latex]\\tan \\left(\\alpha +\\beta \\right)=\\frac{\\sin \\left(\\alpha +\\beta \\right)}{\\cos \\left(\\alpha +\\beta \\right)}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div style=\"text-align: left;\"><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600; text-align: left;\">Use sum and difference formulas for cofunctions<\/span><\/div>\n<\/div>\n<p>Now that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to do the same for their cofunctions. You may recall that if the sum of two positive angles is [latex]\\frac{\\pi }{2}[\/latex], those two angles are complements, and the sum of the two acute angles in a right triangle is [latex]\\frac{\\pi }{2}[\/latex], so they are also complements. In Figure 6, notice that if one of the acute angles is labeled as [latex]\\theta[\/latex], then the other acute angle must be labeled [latex]\\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164045\/CNX_Precalc_Figure_07_02_0072.jpg\" alt=\"Image of a right triangle. The remaining angles are labeled theta and pi\/2 - theta.\" width=\"487\" height=\"268\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6.<\/b> From these relationships, the cofunction identities are formed.<\/p>\n<\/div>\n<p>Notice also that [latex]\\sin \\theta =\\cos \\left(\\frac{\\pi }{2}-\\theta \\right):[\/latex] opposite over hypotenuse. Thus, when two angles are complimentary, we can say that the sine of [latex]\\theta[\/latex] equals the <strong>cofunction<\/strong> of the complement of [latex]\\theta[\/latex]. Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.<span id=\"fs-id2872065\"><br \/>\n<\/span><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Cofunction Identities<\/h3>\n<p>The cofunction identities are summarized in the table below.<\/p>\n<table id=\"Table_07_02_02\" summary=\"Three rows, two columns\/ The table has ordered pairs of these row values: (sin(theta) = cos(pi\/2 - theta), cos(theta) = sin(pi\/2 - theta)), (tan(theta) = cot(pi\/2 - theta), cot(theta) = tan(pi\/2 - theta)), and (sec(theta) = csc(pi\/2 - theta), csc(theta) = sec(pi\/2 - theta)).\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td>[latex]\\sin \\theta =\\cos \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\n<td>[latex]\\cos \\theta =\\sin \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\tan \\theta =\\cot \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\n<td>[latex]\\cot \\theta =\\tan \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sec \\theta =\\csc \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\n<td>[latex]\\csc \\theta =\\sec \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Notice that the formulas in the table may also justified algebraically using the sum and difference formulas. For example, using<\/p>\n<div style=\"text-align: center;\">[latex]\\cos \\left(\\alpha -\\beta \\right)=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta[\/latex],<\/div>\n<p>we can write<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\frac{\\pi }{2}-\\theta \\right)&=\\cos \\frac{\\pi }{2}\\cos \\theta +\\sin \\frac{\\pi }{2}\\sin \\theta \\\\ &=\\left(0\\right)\\cos \\theta +\\left(1\\right)\\sin \\theta \\\\ &=\\sin \\theta\\end{align}[\/latex]<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 7: Finding a Cofunction with the Same Value as the Given Expression<\/h3>\n<p>Write [latex]\\tan \\frac{\\pi }{9}[\/latex] in terms of its cofunction.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q406106\">Show Solution<\/span><\/p>\n<div id=\"q406106\" class=\"hidden-answer\" style=\"display: none\">\n<p>The cofunction of [latex]\\tan \\theta =\\cot \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\frac{\\pi }{9}\\right)&=\\cot \\left(\\frac{\\pi }{2}-\\frac{\\pi }{9}\\right) \\\\& =\\cot \\left(\\frac{9\\pi }{18}-\\frac{2\\pi }{18}\\right) \\\\& =\\cot \\left(\\frac{7\\pi }{18}\\right)\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Write [latex]\\sin \\frac{\\pi }{7}[\/latex] in terms of its cofunction.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q615431\">Show Solution<\/span><\/p>\n<div id=\"q615431\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\cos \\left(\\frac{5\\pi }{14}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm173455\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173455&theme=oea&iframe_resize_id=ohm173455\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use sum and difference formulas to verify identities<\/h2>\n<p>Verifying an identity means demonstrating that the equation holds for all values of the variable. It helps to be very familiar with the identities or to have a list of them accessible while working the problems.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given an identity, verify using sum and difference formulas.<\/h3>\n<ol>\n<li>Begin with the expression on the side of the equal sign that appears most complex. Rewrite that expression until it matches the other side of the equal sign. Occasionally, we might have to alter both sides, but working on only one side is the most efficient.<\/li>\n<li>Look for opportunities to use the sum and difference formulas.<\/li>\n<li>Rewrite sums or differences of quotients as single quotients.<\/li>\n<li>If the process becomes cumbersome, rewrite the expression in terms of sines and cosines.<\/li>\n<\/ol>\n<\/div>\n<div>\n<div>\n<div class=\"textbox shaded\">\n<h3>Example 8: Verifying an Identity Involving Sine<\/h3>\n<p>Verify the identity [latex]\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)=2\\sin \\alpha \\cos \\beta[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q464113\">Show Solution<\/span><\/p>\n<div id=\"q464113\" class=\"hidden-answer\" style=\"display: none\">\n<p>We see that the left side of the equation includes the sines of the sum and the difference of angles.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta\\\\ \\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta \\end{gathered}[\/latex]<\/p>\n<p>We can rewrite each using the sum and difference formulas.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)&=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta +\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta \\\\ &=2\\sin \\alpha \\cos \\beta \\end{align}[\/latex]<\/p>\n<p>We see that the identity is verified.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div>\n<div class=\"textbox shaded\">\n<h3>Example 9: Verifying an Identity Involving Tangent<\/h3>\n<p>Verify the following identity.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\sin \\left(\\alpha -\\beta \\right)}{\\cos \\alpha \\cos \\beta }=\\tan \\alpha -\\tan \\beta[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q605610\">Show Solution<\/span><\/p>\n<div id=\"q605610\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can begin by rewriting the numerator on the left side of the equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\sin \\left(\\alpha -\\beta \\right)}{\\cos \\alpha \\cos \\beta }&=\\frac{\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta } \\\\ &=\\frac{\\sin \\alpha \\cos \\beta}{\\cos \\alpha \\cos \\beta}-\\frac{\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta } && \\text{Rewrite using a common denominator}. \\\\ &=\\frac{\\sin \\alpha }{\\cos \\alpha }-\\frac{\\sin \\beta }{\\cos \\beta }&& \\text{Cancel}. \\\\ &=\\tan \\alpha -\\tan \\beta && \\text{Rewrite in terms of tangent}.\\end{align}[\/latex]<\/p>\n<p>We see that the identity is verified. In many cases, verifying tangent identities can successfully be accomplished by writing the tangent in terms of sine and cosine.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Verify the identity: [latex]\\tan \\left(\\pi -\\theta \\right)=-\\tan \\theta[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q184004\">Show Solution<\/span><\/p>\n<div id=\"q184004\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\pi -\\theta \\right)&=\\frac{\\tan \\left(\\pi \\right)-\\tan \\theta }{1+\\tan \\left(\\pi \\right)\\tan \\theta } \\\\ &=\\frac{0-\\tan \\theta }{1+0\\cdot \\tan \\theta } \\\\ &=-\\tan \\theta \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<h3>Example 10: Using Sum and Difference Formulas to Solve an Application Problem<\/h3>\n<p>Let [latex]{L}_{1}[\/latex] and [latex]{L}_{2}[\/latex] denote two non-vertical intersecting lines, and let [latex]\\theta[\/latex] denote the acute angle between [latex]{L}_{1}[\/latex] and [latex]{L}_{2}[\/latex]. Show that<\/p>\n<p style=\"text-align: center;\">[latex]\\tan \\theta =\\frac{{m}_{2}-{m}_{1}}{1+{m}_{1}{m}_{2}}[\/latex]<\/p>\n<p>where [latex]{m}_{1}[\/latex] and [latex]{m}_{2}[\/latex] are the slopes of [latex]{L}_{1}[\/latex] and [latex]{L}_{2}[\/latex] respectively. (<strong>Hint:<\/strong> Use the fact that [latex]\\tan {\\theta }_{1}={m}_{1}[\/latex] and [latex]\\tan {\\theta }_{2}={m}_{2}[\/latex]. )<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164047\/CNX_Precalc_Figure_07_02_0052.jpg\" alt=\"Diagram of two non-vertical intersecting lines L1 and L2 also intersecting the x-axis. The acute angle formed by the intersection of L1 and L2 is theta. The acute angle formed by L2 and the x-axis is theta 1, and the acute angle formed by the x-axis and L1 is theta 2.\" width=\"487\" height=\"289\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q429859\">Show Solution<\/span><\/p>\n<div id=\"q429859\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using the difference formula for tangent, this problem does not seem as daunting as it might.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\theta &=\\tan \\left({\\theta }_{2}-{\\theta }_{1}\\right) \\\\ &=\\frac{\\tan {\\theta }_{2}-\\tan {\\theta }_{1}}{1+\\tan {\\theta }_{1}\\tan {\\theta }_{2}} \\\\ &=\\frac{{m}_{2}-{m}_{1}}{1+{m}_{1}{m}_{2}}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><span style=\"text-align: center; font-size: 0.9em;\">\u00a0<\/span><\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<div class=\"textbox shaded\">\n<h3>Example 11: Investigating a Guy-wire Problem<\/h3>\n<div style=\"width: 600px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164049\/CNX_Precalc_Figure_07_02_0062.jpg\" alt=\"Two right triangles. Both share the same base, 50 feet. The first has a height of 40 ft and hypotenuse S. The second has height 47 ft and hypotenuse R. The height sides of the triangles are overlapping. There is a B degree angle between R and the base, and an a degree angle between the two hypotenuses within the B degree angle.\" width=\"590\" height=\"322\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8<\/b><\/p>\n<\/div>\n<p>For a climbing wall, a guy-wire [latex]R[\/latex] is attached 47 feet high on a vertical pole. Added support is provided by another guy-wire [latex]S[\/latex] attached 40 feet above ground on the same pole. If the wires are attached to the ground 50 feet from the pole, find the angle [latex]\\alpha[\/latex] between the wires.\u00a0<span id=\"fs-id1386435\"><br \/>\n<\/span><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q48078\">Show Solution<\/span><\/p>\n<div id=\"q48078\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let\u2019s first summarize the information we can gather from the diagram. As only the sides adjacent to the right angle are known, we can use the tangent function. Notice that [latex]\\tan \\beta =\\frac{47}{50}[\/latex], and [latex]\\tan \\left(\\beta -\\alpha \\right)=\\frac{40}{50}=\\frac{4}{5}[\/latex]. We can then use difference formula for tangent.<\/p>\n<p style=\"text-align: center;\">[latex]\\tan \\left(\\beta -\\alpha \\right)=\\frac{\\tan \\beta -\\tan \\alpha }{1+\\tan \\beta \\tan \\alpha }[\/latex]<\/p>\n<p>Now, substituting the values we know into the formula, we have<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\frac{4}{5}=\\frac{\\frac{47}{50}-\\tan \\alpha }{1+\\frac{47}{50}\\tan \\alpha } \\\\ 4\\left(1+\\frac{47}{50}\\tan \\alpha \\right)=5\\left(\\frac{47}{50}-\\tan \\alpha \\right) \\end{gathered}[\/latex]<\/p>\n<p>Use the distributive property, and then simplify the functions.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}4\\left(1\\right)+4\\left(\\frac{47}{50}\\right)\\tan \\alpha =5\\left(\\frac{47}{50}\\right)-5\\tan \\alpha\\\\ 4+3.76\\tan \\alpha =4.7 - 5\\tan \\alpha \\\\ 5\\tan \\alpha +3.76\\tan \\alpha =0.7\\\\ 8.76\\tan \\alpha =0.7\\\\ \\tan \\alpha \\approx 0.07991 \\\\ {\\tan }^{-1}\\left(0.07991\\right)\\approx .079741 \\end{gathered}[\/latex]<\/p>\n<p>Now we can calculate the angle in degrees.<\/p>\n<p style=\"text-align: center;\">[latex]\\alpha \\approx 0.079741\\left(\\frac{180}{\\pi }\\right)\\approx {4.57}^{\\circ }[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Occasionally, when an application appears that includes a right triangle, we may think that solving is a matter of applying the Pythagorean Theorem. That may be partially true, but it depends on what the problem is asking and what information is given.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Key Equations<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<table>\n<tbody>\n<tr>\n<td>Sum Formula for Cosine<\/td>\n<td>[latex]\\cos \\left(\\alpha +\\beta \\right)=\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Difference Formula for Cosine<\/td>\n<td>[latex]\\cos \\left(\\alpha -\\beta \\right)=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Sum Formula for Sine<\/td>\n<td>[latex]\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Difference Formula for Sine<\/td>\n<td>[latex]\\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Sum Formula for Tangent<\/td>\n<td>[latex]\\tan \\left(\\alpha +\\beta \\right)=\\frac{\\tan \\alpha +\\tan \\beta }{1-\\tan \\alpha \\tan \\beta }[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Difference Formula for Tangent<\/td>\n<td>[latex]\\tan \\left(\\alpha -\\beta \\right)=\\frac{\\tan \\alpha -\\tan \\beta }{1+\\tan \\alpha \\tan \\beta }[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Cofunction identities<\/td>\n<td>[latex]\\sin \\theta =\\cos \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/p>\n<p>[latex]\\cos \\theta =\\sin \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/p>\n<p>[latex]\\tan \\theta =\\cot \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/p>\n<p>[latex]\\cot \\theta =\\tan \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/p>\n<p>[latex]\\sec \\theta =\\csc \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/p>\n<p>[latex]\\csc \\theta =\\sec \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>The sum formula for cosines states that the cosine of the sum of two angles equals the product of the cosines of the angles minus the product of the sines of the angles. The difference formula for cosines states that the cosine of the difference of two angles equals the product of the cosines of the angles plus the product of the sines of the angles.<\/li>\n<li>The sum and difference formulas can be used to find the exact values of the sine, cosine, or tangent of an angle.<\/li>\n<li>The sum formula for sines states that the sine of the sum of two angles equals the product of the sine of the first angle and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second angle. The difference formula for sines states that the sine of the difference of two angles equals the product of the sine of the first angle and cosine of the second angle minus the product of the cosine of the first angle and the sine of the second angle.<\/li>\n<li>The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions.<\/li>\n<li>The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by 1 minus the product of the tangents of the angles. The difference formula for tangent states that the tangent of the difference of two angles equals the difference of the tangents of the angles divided by 1 plus the product of the tangents of the angles.<\/li>\n<li>The Pythagorean Theorem along with the sum and difference formulas can be used to find multiple sums and differences of angles.<\/li>\n<li>The cofunction identities apply to complementary angles and pairs of reciprocal functions.<\/li>\n<li>Sum and difference formulas are useful in verifying identities.<\/li>\n<li>Application problems are often easier to solve by using sum and difference formulas.<\/li>\n<\/ul>\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-193\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <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":23485,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"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-193","chapter","type-chapter","status-publish","hentry"],"part":191,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/193","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/users\/23485"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/193\/revisions"}],"predecessor-version":[{"id":671,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/193\/revisions\/671"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/parts\/191"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/193\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/media?parent=193"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=193"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/contributor?post=193"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/license?post=193"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}