{"id":17057,"date":"2020-04-13T19:56:46","date_gmt":"2020-04-13T19:56:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/?post_type=chapter&p=17057"},"modified":"2020-05-21T05:26:29","modified_gmt":"2020-05-21T05:26:29","slug":"chapter-5-solutions-to-odd-numbered-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/chapter\/chapter-5-solutions-to-odd-numbered-problems\/","title":{"raw":"Chapter 4 Solutions to Odd-Numbered Problems","rendered":"Chapter 4 Solutions to Odd-Numbered Problems"},"content":{"raw":"

Section 4.1 Solutions<\/h2>\r\n1.\r\n\"Graph\r\n\r\n3.\u00a0Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.\r\n\r\n5.\u00a0Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.\r\n\r\n7.\r\n\"Graph\r\n\r\n9.\r\n\"Graph\r\n\r\n11.\r\n\"Graph\r\n\r\n13.\r\n\"Graph\r\n\r\n15.\r\n\"Graph\r\n\r\n17.\u00a0240\u00b0\r\n\"Graph<\/span>\r\n\r\n19.\u00a0[latex]\\frac{4\\pi }{3}[\/latex]\r\n\"Graph<\/span>\r\n\r\n21.\u00a0[latex]\\frac{2\\pi }{3}[\/latex]\r\n\"Graph<\/span>\r\n\r\n23.\u00a0[latex]\\frac{7\\pi }{2}\\approx 11.00{\\text{ in}}^{2}[\/latex]\r\n\r\n25.\u00a0[latex]\\frac{9\\pi }{5}\\approx 5.65{\\text{ cm}}^{2}[\/latex]\r\n\r\n27.\u00a020\u00b0\r\n\r\n29.\u00a060\u00b0\r\n\r\n31.\u00a0\u221275\u00b0\r\n\r\n33.\u00a0[latex]\\frac{\\pi }{2}[\/latex] radians\r\n\r\n35.\u00a0[latex]-3\\pi [\/latex] radians\r\n\r\n37.\u00a0[latex]\\pi [\/latex] radians\r\n\r\n39.\u00a0[latex]\\frac{5\\pi }{6}[\/latex] radians\r\n\r\n41. [latex]154.795^\\circ[\/latex]\r\n\r\n43. [latex]30.23^\\circ[\/latex]\r\n\r\n45. [latex]2^\\circ 55' 21''[\/latex]\r\n\r\n47. [latex]36^\\circ 52' 12''[\/latex]\r\n\r\n49.\u00a0[latex]\\frac{5.02\\pi }{3}\\approx 5.26[\/latex] miles\r\n\r\n51.\u00a0[latex]\\frac{25\\pi }{9}\\approx 8.73[\/latex] centimeters\r\n\r\n53.\u00a0[latex]\\frac{21\\pi }{10}\\approx 6.60[\/latex] meters\r\n\r\n55.\u00a0104.7198 cm2\r\n\r\n57.\u00a00.7697 in2\r\n\r\n59.\u00a0250\u00b0\r\n\r\n61.\u00a0320\u00b0\r\n\r\n63.\u00a0[latex]\\frac{4\\pi }{3}[\/latex]\r\n\r\n65.\u00a0[latex]\\frac{8\\pi }{9}[\/latex]\r\n\r\n67.\u00a01320 rad 210.085 RPM\r\n\r\n69.\u00a07 in.\/s, 4.77 RPM, 28.65 deg\/s\r\n\r\n71.\u00a0[latex]1,809,557.37\\text{ mm\/min}=30.16\\text{ m\/s}[\/latex]\r\n\r\n73.\u00a0[latex]5.76[\/latex]\u00a0miles\r\n\r\n75.\u00a0[latex]120^\\circ [\/latex]\r\n\r\n77.\u00a0794 miles per hour\r\n\r\n79.\u00a02,234 miles per hour\r\n\r\n81.\u00a011.5 inches\r\n

Section 4.2 Solutions<\/h2>\r\n1. The unit circle is a circle of radius 1 centered at the origin.\r\n\r\n3.\u00a0Yes, when the reference angle is [latex]\\frac{\\pi }{4}[\/latex] and the terminal side of the angle is in quadrants I and III. Thus, at [latex]x=\\frac{\\pi }{4},\\frac{5\\pi }{4}[\/latex], the sine and cosine values are equal.\r\n\r\n5.\u00a0Substitute the sine of the angle in for [latex]y[\/latex] in the Pythagorean Theorem [latex]{x}^{2}+{y}^{2}=1[\/latex]. Solve for [latex]x[\/latex] and take the negative solution.\r\n\r\n7. I\r\n\r\n9. IV\r\n\r\n11.\u00a0[latex]\\frac{\\sqrt{3}}{2}\\text{ , }\\frac{2\\sqrt{3}}{3}[\/latex]\r\n\r\n13.\u00a0[latex]\\frac{1}{2}\\text{ , }2[\/latex]\r\n\r\n15.\u00a0[latex]\\frac{\\sqrt{2}}{2}\\text{ , }\\sqrt{3}[\/latex]\r\n\r\n17. [latex]0{ , }\\sqrt{2}[\/latex]\r\n\r\n19. [latex]-1\\text{ , }0[\/latex]\r\n\r\n21. [latex]1\\text{ , }0[\/latex]\r\n\r\n23.\u00a0[latex]\\frac{\\sqrt{77}}{9}[\/latex]\r\n\r\n25.\u00a0[latex]-\\frac{\\sqrt{15}}{4}[\/latex]\r\n\r\n27.\u00a0[latex]\\sin t=\\frac{1}{2},\\csc t=2,\\cos t=-\\frac{\\sqrt{3}}{2},\\sec t=-\\frac{2\\sqrt{3}}{3},\\tan t=-\\frac{\\sqrt{3}}{3},\\cot t=-\\sqrt{3} [\/latex]\r\n\r\n29.\u00a0[latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\csc t=-\\sqrt{2},\\cos t=-\\frac{\\sqrt{2}}{2},\\sec t=-\\sqrt{2},\\tan t=1,\\cot t=1[\/latex]\r\n\r\n31.\u00a0[latex]\\sin t=\\frac{\\sqrt{3}}{2},\\csc t=\\frac{2\\sqrt{3}}{3},\\cos t=-\\frac{1}{2},\\sec t=-2,\\tan t=\\sqrt{3},\\cot t=\\frac{\\sqrt{3}}{3}[\/latex]\r\n\r\n33.\u00a0[latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\csc t=-\\sqrt{2},\\cos t=\\frac{\\sqrt{2}}{2},\\sec t=\\sqrt{2},\\tan t=-1,\\cot t=1[\/latex]\r\n\r\n35.\u00a0[latex]\\sin t=0,\\csc t=\\varnothing,\\cos t=-1,\\sec t=-1,\\tan t=0,\\cot t=\\varnothing[\/latex]\r\n\r\n37.\u00a0[latex]\\sin t=-0.596,\\csc t=-1.679,\\cos t=0.803,\\sec t=1.245,\\tan t=-0.742,\\cot t=-1.347[\/latex]\r\n\r\n39.\u00a0\u22120.1736\r\n\r\n41.\u00a00.9511\r\n\r\n43.\u00a0\u22120.7071\r\n\r\n45.\u00a0\u22120.1392\r\n\r\n47.\u00a0\u22120.7660\r\n\r\n49.\u00a0\u20130.228\r\n\r\n51.\u00a0\u20132.414\r\n\r\n53.\u00a01.556\r\n\r\n55.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]\r\n\r\n57.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]\r\n\r\n59. 0\r\n\r\n61.\u00a0[latex]\\cos\\left(6t\\right)-\\sin\\left(9t\\right)[\/latex]\r\n\r\n63.\u00a0even\r\n\r\n65.\u00a0even\r\n\r\n67.\u00a013.77 hours, period: [latex]1000\\pi [\/latex]\r\n\r\n69.\u00a07.73 inches\r\n

Section 4.3 Solutions<\/h2>\r\n1.\r\n\"A\r\n\r\n3. The tangent of an angle is the ratio of the opposite side to the adjacent side.\r\n\r\n5.\u00a0For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.\r\n\r\n7. [latex]\\frac{\\sqrt{2}-4}{4}[\/latex]\r\n\r\n9. 5\r\n\r\n11.\u00a0[latex]\\frac{\\pi }{6}[\/latex]\r\n\r\n13.\u00a0[latex]\\frac{\\pi }{4}[\/latex]\r\n\r\n15.\u00a0[latex]b=\\frac{20\\sqrt{3}}{3},c=\\frac{40\\sqrt{3}}{3}[\/latex]\r\n\r\n17.\u00a0[latex]a=10,000,c=10,000.5[\/latex]\r\n\r\n19.\u00a0[latex]b=\\frac{5\\sqrt{3}}{3},c=\\frac{10\\sqrt{3}}{3}[\/latex]\r\n\r\n21.\u00a0[latex]\\frac{5\\sqrt{29}}{29}[\/latex]\r\n\r\n23.\u00a0[latex]\\frac{5}{2}[\/latex]\r\n\r\n25.\u00a0[latex]\\frac{\\sqrt{29}}{2}[\/latex]\r\n\r\n27.\u00a0[latex]\\frac{5\\sqrt{41}}{41}[\/latex]\r\n\r\n29.\u00a0[latex]\\frac{5}{4}[\/latex]\r\n\r\n31.\u00a0[latex]\\frac{\\sqrt{41}}{4}[\/latex]\r\n\r\n33.\u00a0[latex]c=14, b=7\\sqrt{3}[\/latex]\r\n\r\n35.\u00a0[latex]a=15, b=15[\/latex]\r\n\r\n37.\u00a0[latex]b=9.9970, c=12.2041[\/latex]\r\n\r\n39.\u00a0[latex]a=2.0838, b=11.8177[\/latex]\r\n\r\n41.\u00a0[latex]a=55.9808,c=57.9555[\/latex]\r\n\r\n43.\u00a0[latex]a=46.6790,b=17.9184[\/latex]\r\n\r\n45.\u00a0[latex]a=16.4662,c=16.8341[\/latex]\r\n\r\n47.\u00a0188.3159\r\n\r\n49.\u00a0200.6737\r\n\r\n51.\u00a0498.3471 ft\r\n\r\n53.\u00a01060.09 ft\r\n\r\n55.\u00a027.372 ft\r\n\r\n57.\u00a022.6506 ft\r\n\r\n59.\u00a0368.7633 ft\r\n\r\n61. [latex]S 29.05^\\circ W[\/latex]\r\n\r\n63. East: 13.49 inches, North: 33.38 inches\r\n\r\n65. [latex]18.3^\\circ[\/latex]\r\n

Section 4.4 Solutions<\/h2>\r\n1.\u00a0Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, [latex]t[\/latex], formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis.\r\n\r\n3. The sine values are equal.\r\n\r\n5.\u00a0[latex]60^\\circ [\/latex]\r\n\r\n7.\u00a0[latex]80^\\circ [\/latex]\r\n\r\n9.\u00a0[latex]45^\\circ [\/latex]\r\n\r\n11.\u00a0[latex]\\frac{\\pi }{3}[\/latex]\r\n\r\n13.\u00a0[latex]\\frac{\\pi }{3}[\/latex]\r\n\r\n15.\u00a0[latex]\\frac{\\pi }{8}[\/latex]\r\n\r\n17.\u00a0[latex]60^\\circ [\/latex], Quadrant IV, [latex]\\text{sin}\\left(300^\\circ \\right)=-\\frac{\\sqrt{3}}{2},\\cos \\left(300^\\circ \\right)=\\frac{1}{2}[\/latex]\r\n\r\n19.\u00a0[latex]45^\\circ [\/latex], Quadrant II, [latex]\\text{sin}\\left(135^\\circ \\right)=\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\cos \\left(135^\\circ \\right)=-\\frac{\\sqrt{2}}{2}[\/latex]\r\n\r\n21.\u00a0[latex]60^\\circ [\/latex], Quadrant II, [latex]\\text{sin}\\left(480^\\circ \\right)=\\frac{\\sqrt{3}}{2}[\/latex], [latex]\\cos \\left(480^\\circ \\right)=-\\frac{1}{2}[\/latex]\r\n\r\n23.\u00a0[latex]30^\\circ [\/latex], Quadrant II, [latex]\\text{sin}\\left(-210^\\circ \\right)=\\frac{1}{2}[\/latex], [latex]\\cos \\left(-210^\\circ \\right)=-\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n25.\u00a0[latex]\\frac{\\pi }{6}[\/latex], Quadrant III, [latex]\\text{sin}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{1}{2}[\/latex], [latex]\\text{cos}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n27.\u00a0[latex]\\frac{\\pi }{4}[\/latex], Quadrant II, [latex]\\text{sin}\\left(\\frac{3\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\cos \\left(\\frac{4\\pi }{3}\\right)=-\\frac{\\sqrt[]{2}}{2}[\/latex]\r\n\r\n29.\u00a0[latex]\\frac{\\pi }{3}[\/latex], Quadrant II, [latex]\\text{sin}\\left(\\frac{2\\pi }{3}\\right)=\\frac{\\sqrt{3}}{2}[\/latex], [latex]\\cos \\left(\\frac{2\\pi }{3}\\right)=-\\frac{1}{2}[\/latex]\r\n\r\n31.\u00a0[latex]\\frac{\\pi }{4}[\/latex], Quadrant IV, [latex]\\text{sin}\\left(\\frac{-9\\pi }{4}\\right)=-\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\text{cos}\\left(\\frac{-9\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2}[\/latex]\r\n\r\n33.\u00a0[latex]\\frac{\\pi }{6}[\/latex], Quadrant III, [latex]\\text{sec}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{2\\sqrt{3}}{3}[\/latex]\r\n\r\n35.\u00a0[latex]\\frac{\\pi }{6}[\/latex], Quadrant I, [latex]\\text{cot}\\left(\\frac{13\\pi }{6}\\right)=\\sqrt{3}[\/latex]\r\n\r\n37.\u00a0[latex]\\frac{\\pi }{4}[\/latex], Quadrant II, [latex]\\text{sec}\\left(\\frac{3\\pi }{4}\\right)=-\\sqrt{2}[\/latex]\r\n\r\n39.\u00a0[latex]\\frac{\\pi }{4}[\/latex], Quadrant IV, [latex]\\text{cot}\\left(\\frac{11\\pi }{4}\\right)=-1[\/latex]\r\n\r\n41.\u00a0[latex]\\frac{\\pi }{3}[\/latex], Quadrant III, [latex]\\text{sec}\\left(-\\frac{2\\pi }{3}\\right)=-2[\/latex]\r\n\r\n43. [latex]\\frac{\\pi }{3}[\/latex], Quadrant IV, [latex]\\text{cot}\\left(-\\frac{7\\pi }{3}\\right)=-\\frac{\\sqrt{3}}{3}[\/latex]\r\n\r\n45.\u00a0[latex]\\text{ }60^\\circ[\/latex], Quadrant IV, [latex]\\text{sec}\\left(300^\\circ\\right)=2[\/latex]\r\n\r\n47.\u00a0[latex]\\text{ }60^\\circ[\/latex], Quadrant III, [latex]\\text{cot}\\left(600^\\circ\\right)=\\frac{\\sqrt{3}}{3}[\/latex]\r\n\r\n49.\u00a0[latex]\\text{ }30^\\circ[\/latex], Quadrant II, [latex]\\text{sec}\\left(-210^\\circ\\right)=-\\frac{2\\sqrt{3}}{3}[\/latex]\r\n\r\n51.\u00a0[latex]\\text{ }45^\\circ[\/latex], Quadrant IV, [latex]\\text{cot}\\left(-405^\\circ\\right)=-1[\/latex]\r\n\r\n53.\u00a0If [latex]\\text{ }\\sin t=-\\frac{2\\sqrt{2}}{3},\\sec t=-3,\\csc t=-\\frac{3\\sqrt{2}}{4},\\tan t=2\\sqrt{2},\\cot t=\\frac{\\sqrt{2}}{4}[\/latex]\r\n\r\n55.\u00a0[latex]\\text{ }\\sec t=2,\\csc t=\\frac{2\\sqrt{3}}{3},\\tan t=\\sqrt{3},\\cot t=\\frac{\\sqrt{3}}{3}[\/latex]\r\n\r\n57.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]\r\n\r\n59.\u00a0[latex]-\\frac{\\sqrt{6}}{4}[\/latex]\r\n\r\n61.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]\r\n\r\n63.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]\r\n\r\n65. 0\r\n

Section 4.5 Solutions<\/h2>\r\n1. The sine and cosine functions have the property that [latex]f(x+P)=f(x)[\/latex] for a certain P<\/em>. This means that the function values repeat for every P<\/em> units on the x-axis.\r\n\r\n3. The absolute value of the constant A<\/em> (amplitude) increases the total range and the constant D<\/em> (vertical shift) shifts the graph vertically.\r\n\r\n5. At the point where the terminal side of t<\/em> intersects the unit circle, you can determine that the sin t<\/em> equals the y<\/em>-coordinate of the point.\r\n\r\n7. amplitude: [latex]\\frac{2}{3}[\/latex]; period: 2\u03c0; midline: [latex]y=0[\/latex]; maximum: [latex]y=23[\/latex] occurs at [latex]x=0[\/latex]; minimum: [latex]y=\u221223[\/latex] occurs at [latex]x=\\pi[\/latex]; for one period, the graph starts at 0 and ends at 2\u03c0\r\n\"A\r\n\r\n9. amplitude: 4; period: 2\u03c0; midline: [latex]y=0[\/latex]; maximum [latex]y=4[\/latex] occurs at [latex]x=\\frac{\\pi}{2}[\/latex]; minimum: [latex]y=\u22124[\/latex] occurs at [latex]x=\\frac{3\\pi}{2}[\/latex]; one full period occurs from [latex]x=0[\/latex] to [latex]x=2\u03c0[\/latex]\r\n\"A\r\n\r\n11. amplitude: 1; period: \u03c0; midline: y=0; maximum: y=1 occurs at [latex]x=\\pi[\/latex]; minimum: [latex]y=\u22121[\/latex] occurs at [latex]x=\\frac{\\pi}{2}[\/latex]; one full period is graphed from [latex]x=0[\/latex] to [latex]x=\\pi[\/latex]\r\n\"A\r\n\r\n13. amplitude: 4; period: 2; midline: [latex]y=0[\/latex]; maximum: [latex]y=4[\/latex] occurs at [latex]x=0[\/latex]; minimum: [latex]y=\u22124[\/latex] occurs at [latex]x=1[\/latex]\r\n\"A\r\n\r\n15. amplitude: 3; period: [latex]\\frac{\\pi}{4}[\/latex]; midline: [latex]y=5[\/latex]; maximum: [latex]y=8[\/latex] occurs at [latex]x=0.12[\/latex]; minimum: [latex]y=2[\/latex] occurs at [latex]x=0.516[\/latex]; horizontal shift: \u22124; vertical translation 5; one period occurs from [latex]x=0[\/latex] to [latex]x=\\frac{\\pi}{4}[\/latex]\r\n\"A\r\n\r\n17. amplitude: 5; period: [latex]\\frac{2\\pi}{5}; midline: [latex]y=\u22122[\/latex]; maximum: [latex]y=3[\/latex] occurs at [latex]x=0.08[\/latex]; minimum: [latex]y=\u22127[\/latex] occurs at [latex]x=0.71[\/latex]; phase shift:\u22124; vertical translation:\u22122; one full period can be graphed on [latex]x=0[\/latex] to [latex]x=\\frac{2\\pi}{5}[\/latex]\r\n\"A\r\n\r\n19. amplitude: 1; period: 2\u03c0; midline: y=1; maximum:[latex]y=2[\/latex] occurs at [latex]x=2.09[\/latex]; maximum:[latex]y=2[\/latex] occurs at[latex]t=2.09[\/latex]; minimum:[latex]y=0[\/latex] occurs at [latex]t=5.24[\/latex]; phase shift: [latex]\u2212\\frac{\\pi}{3}[\/latex]; vertical translation: 1; one full period is from [latex]t=0[\/latex] to [latex]t=2\u03c0[\/latex]\r\n\"A\r\n\r\n21. amplitude: 1; period: 4\u03c0; midline: [latex]y=0[\/latex]; maximum: [latex]y=1[\/latex] occurs at [latex]t=11.52[\/latex]; minimum: [latex]y=\u22121[\/latex] occurs at [latex]t=5.24[\/latex]; phase shift: \u2212[latex]\\frac{10\\pi}{3}[\/latex]; vertical shift: 0\r\n\"A\r\n\r\n23. amplitude: 2; midline: [latex]y=\u22123[\/latex]; period: 4; equation: [latex]f(x)=2\\sin\\left(\\frac{\\pi}{2}x\\right)\u22123[\/latex]\r\n\r\n25. amplitude: 2; period: 5; midline: [latex]y=3[\/latex]; equation: [latex]f(x)=\u22122\\cos\\left(\\frac{2\\pi}{5}x\\right)+3[\/latex]\r\n\r\n27. amplitude: 4; period: 2; midline: [latex]y=0[\/latex]; equation: [latex]f(x)=\u22124\\cos\\left(\\pi\\left(x\u2212\\frac{\\pi}{2}\\right)\\right)[\/latex]\r\n\r\n29. amplitude: 2; period: 2; midline [latex]y=1[\/latex]; equation: [latex]f(x)=2\\cos\\left(\\frac{\\pi}{x}\\right)+1[\/latex]\r\n\r\n31.\u00a0[latex]\\frac{\\pi}{6},\\frac{5\\pi}{6}[\/latex]\r\n\r\n33.\u00a0[latex]\\frac{\\pi}{4},\\frac{3\\pi}{4}[\/latex]\r\n\r\n35.\u00a0[latex]\\frac{3\\pi}{2}[\/latex]\r\n\r\n37.\u00a0[latex]\\frac{\\pi}{2},\\frac{3\\pi}{2}[\/latex]\r\n\r\n39.\u00a0[latex]\\frac{\\pi}{2},\\frac{3\\pi}{2}[\/latex]\r\n\r\n41.\u00a0[latex]\\frac{\\pi}{6},\\frac{11\\pi}{6}[\/latex]\r\n\r\n43. The graph appears linear. The linear functions dominate the shape of the graph for large values of x.\r\n\"A\r\n\r\n45. The graph is symmetric with respect to the y<\/em>-axis and there is no amplitude because the function is not periodic.\r\n\"A\r\n\r\n47.\r\na. Amplitude: 12.5; period: 10; midline: [latex]y=13.5[\/latex];\r\nb. [latex]h(t)=12.5\\sin\\left(\\frac{\\pi}{5}\\left(t\u22122.5\\right)\\right)+13.5;[\/latex]\r\nc. 26 ft\r\n

Section 4.6 Solutions<\/h2>\r\n1. \u00a0Since [latex]y=\\csc x[\/latex] is the reciprocal function of [latex]y=\\sin x[\/latex], you can plot the reciprocal of the coordinates on the graph of [latex]y=\\sin x[\/latex] to obtain the y<\/em>-coordinates of [latex]y=\\csc x[\/latex]. The x<\/em>-intercepts of the graph [latex]y=\\sin x[\/latex] are the vertical asymptotes for the graph of [latex]y=\\csc x[\/latex].\r\n\r\n3.\u00a0Answers will vary. Using the unit circle, one can show that [latex]\\tan(x+\\pi)=\\tan x[\/latex].\r\n\r\n5.\u00a0The period is the same: 2\u03c0.\r\n\r\n7. IV\r\n\r\n9. III\r\n\r\n11. period: 8; horizontal shift: 1 unit to left\r\n\r\n13. 1.5\r\n\r\n15. 5\r\n\r\n17. [latex]\u2212\\cot x\\cos x\u2212\\sin x[\/latex]\r\n\r\n19.\u00a0stretching factor: 2; period: [latex]\\frac{\\pi}{4}[\/latex]; asymptotes: [latex]x=\\frac{1}{4}\\left(\\frac{\\pi}{2}+\\pi k\\right)+8[\/latex], where k<\/em> is an integer\r\n\"A\r\n\r\n21.\u00a0stretching factor: 6; period: 6; asymptotes: [latex]x=3k[\/latex], where k<\/em> is an integer\r\n\"A\r\n\r\n23.\u00a0stretching factor: 1; period: \u03c0; asymptotes: [latex]x=\u03c0k[\/latex], where k<\/em> is an integer\r\n\"A\r\n\r\n25.\u00a0Stretching factor: 1; period: \u03c0; asymptotes: [latex]x=\\frac{\\pi}{4}+{\\pi}k[\/latex], where k<\/em> is an integer\r\n\"A\r\n\r\n27.\u00a0stretching factor: 2; period: 2\u03c0; asymptotes: [latex]x=\u03c0k[\/latex], where k<\/em> is an integer\r\n\"A\r\n\r\n29.\u00a0stretching factor: 4; period: [latex]\\frac{2\\pi}{3}[\/latex]; asymptotes: [latex]x=\\frac{\\pi}{6}k[\/latex], where k<\/em> is an odd integer\r\n\"A\r\n\r\n31.\u00a0stretching factor: 7; period: [latex]\\frac{2\\pi}{5}[\/latex]; asymptotes: [latex]x=\\frac{\\pi}{10}k[\/latex], where k<\/em> is an odd integer\r\n\"A\r\n\r\n33.\u00a0stretching factor: 2; period: 2\u03c0; asymptotes: [latex]x=\u2212\\frac{\\pi}{4}+\\pi k[\/latex], where k<\/em> is an integer\r\n\"A\r\n\r\n35.\u00a0stretching factor: [latex]\\frac{7}{5}[\/latex]; period: 2\u03c0; asymptotes: [latex]x=\\frac{\\pi}{4}+\\pi[\/latex]k<\/em>, where k<\/em> is an integer\r\n\"A\r\n\r\n37. [latex]y=\\tan\\left(3\\left(x\u2212\\frac{\\pi}{4}\\right)\\right)+2[\/latex]\r\n\"A\r\n\r\n39. [latex]f(x)=\\csc(2x)[\/latex]\r\n\r\n41. [latex]f(x)=\\csc(4x)[\/latex]\r\n\r\n43. [latex]f(x)=2\\csc x[\/latex]\r\n\r\n45. [latex]f(x)=\\frac{1}{2}\\tan(100\\pi x)[\/latex]\r\n\r\nFor the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input [latex]\\csc x[\/latex] as [latex]\\frac{1}{\\sin x}[\/latex].\r\n\r\n46. [latex]f(x)=|\\csc(x)|[\/latex]\r\n\r\n47. [latex]f(x)=|\\cot(x)|[\/latex]\r\n\r\n48. [latex]f(x)=2^{\\csc(x)}[\/latex]\r\n\r\n49. [latex]f(x)=\\frac{\\csc(x)}{\\sec(x)}[\/latex]\r\n\r\n51.\r\n\"A\r\n\r\n53.\r\n\"A\r\n\r\n55. a. [latex](\u2212\\frac{\\pi}{2}\\text{,}\\frac{\\pi}{2})[\/latex];\r\nb.\r\n\"A\r\nc. [latex]x=\u2212\\frac{\\pi}{2}[\/latex] and [latex]x=\\frac{\\pi}{2}[\/latex]; the distance grows without bound as |x<\/em>| approaches [latex]\\frac{\\pi}{2}[\/latex]\u2014i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;\r\nd. 3; when [latex]x=\u2212\\frac{\\pi}{3}[\/latex], the boat is 3 km away;\r\ne. 1.73; when [latex]x=\\frac{\\pi}{6}[\/latex], the boat is about 1.73 km away;\r\nf. 1.5 km; when [latex]x=0[\/latex].\r\n\r\n57. a. [latex]h(x)=2\\tan\\left(\\frac{\\pi}{120}x\\right)[\/latex];\r\nb.\r\n\"An\r\nc. [latex]h(0)=0:[\/latex] after 0 seconds, the rocket is 0 mi above the ground; [latex]h(30)=2:[\/latex] after 30 seconds, the rockets is 2 mi high;\r\nd.\u00a0As x<\/em> approaches 60 seconds, the values of [latex]h(x)[\/latex] grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.\r\n

Section 4.7 Solutions<\/h2>\r\n1.\u00a0The function [latex]y=\\sin x[\/latex] is one-to-one on [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex]; thus, this interval is the range of the inverse function of [latex]y=\\sin x\\text{, }f\\left(x\\right)=\\sin^{\u22121}x[\/latex]. The function [latex]y=\\cos x[\/latex] is one-to-one on [0,\u03c0]; thus, this interval is the range of the inverse function of [latex]y=\\cos x\\text{, }f(x)=\\cos^{\u22121}x[\/latex].\r\n\r\n3. [latex]\\frac{\\pi}{6}[\/latex] is the radian measure of an angle between [latex]\u2212\\frac{\\pi}{2}[\/latex] and [latex]\\frac{\\pi}{2}[\/latex] whose sine is 0.5.\r\n\r\n5.\u00a0In order for any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex] so that it is one-to-one and possesses an inverse.\r\n\r\n7.\u00a0True. The angle, [latex]\\theta_{1}[\/latex] that equals [latex]\\arccos(\u2212x)\\text{, }x\\text{>}0[\/latex], will be a second quadrant angle with reference angle,\u00a0[latex]\\theta_{2}[\/latex], where [latex]\\theta_{2}[\/latex] equals [latex]\\arccos x\\text{, }x\\text{>}0[\/latex]. Since [latex]\\theta_{2}[\/latex] is the reference angle for [latex]\\theta_{1}[\/latex], [latex]\\theta_{2}=\\pi(\u2212x)=\\pi\u2212\\arccos x[\/latex]\r\n\r\n9. [latex]\u2212\\frac{\\pi}{6}[\/latex]\r\n\r\n11. [latex]\\frac{3\\pi}{4}[\/latex]\r\n\r\n13.\u00a0[latex]\u2212\\frac{\\pi}{3}[\/latex]\r\n\r\n15. [latex]\\frac{\\pi}{3}[\/latex]\r\n\r\n17. 1.98\r\n\r\n19. 0.93\r\n\r\n21. 1.41\r\n\r\n23. 0.56 radians\r\n\r\n25. 0\r\n\r\n27. 0.71\r\n\r\n29.\u00a0\u22120.71\r\n\r\n31. [latex]\u2212\\frac{\\pi}{4}[\/latex]\r\n\r\n33. 0.8\r\n\r\n35. [latex]\\frac{5}{13}[\/latex]\r\n\r\n37. [latex]\\frac{x}{\\sqrt{1\u2212x^{2}}}[\/latex]\r\n\r\n39. [latex]\\frac{\\sqrt{x^{2}\u22121}}{x}[\/latex]\r\n\r\n41. [latex]\\frac{2x}{\\sqrt{4x^{2}+1}}[\/latex]\r\n\r\n43. [latex]\\frac{\\sqrt{2x+1}}{x+1}[\/latex]\r\n\r\n45. [latex]\\frac{\\sqrt{2x+1}}{x}[\/latex]\r\n\r\n47.\u00a0[latex]\\frac{x}{\\sqrt{2x+1}}[\/latex]\r\n\r\n49.\u00a0domain [\u22121,1]; range [0,\u03c0]\r\n\"A\r\n\r\n51. approximately [latex]x=0.00[\/latex]\r\n\r\n53. 0.395 radians\r\n\r\n55. 1.11 radians\r\n\r\n57. 1.25 radians\r\n\r\n59. 0.405 radians\r\n\r\n61. No. The angle the ladder makes with the horizontal is 60 degrees.","rendered":"

Section 4.1 Solutions<\/h2>\n

1.
\n\"Graph<\/p>\n

3.\u00a0Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.<\/p>\n

5.\u00a0Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.<\/p>\n

7.
\n\"Graph<\/p>\n

9.
\n\"Graph<\/p>\n

11.
\n\"Graph<\/p>\n

13.
\n\"Graph<\/p>\n

15.
\n\"Graph<\/p>\n

17.\u00a0240\u00b0
\n\"Graph<\/span><\/p>\n

19.\u00a0[latex]\\frac{4\\pi }{3}[\/latex]
\n\"Graph<\/span><\/p>\n

21.\u00a0[latex]\\frac{2\\pi }{3}[\/latex]
\n\"Graph<\/span><\/p>\n

23.\u00a0[latex]\\frac{7\\pi }{2}\\approx 11.00{\\text{ in}}^{2}[\/latex]<\/p>\n

25.\u00a0[latex]\\frac{9\\pi }{5}\\approx 5.65{\\text{ cm}}^{2}[\/latex]<\/p>\n

27.\u00a020\u00b0<\/p>\n

29.\u00a060\u00b0<\/p>\n

31.\u00a0\u221275\u00b0<\/p>\n

33.\u00a0[latex]\\frac{\\pi }{2}[\/latex] radians<\/p>\n

35.\u00a0[latex]-3\\pi[\/latex] radians<\/p>\n

37.\u00a0[latex]\\pi[\/latex] radians<\/p>\n

39.\u00a0[latex]\\frac{5\\pi }{6}[\/latex] radians<\/p>\n

41. [latex]154.795^\\circ[\/latex]<\/p>\n

43. [latex]30.23^\\circ[\/latex]<\/p>\n

45. [latex]2^\\circ 55' 21''[\/latex]<\/p>\n

47. [latex]36^\\circ 52' 12''[\/latex]<\/p>\n

49.\u00a0[latex]\\frac{5.02\\pi }{3}\\approx 5.26[\/latex] miles<\/p>\n

51.\u00a0[latex]\\frac{25\\pi }{9}\\approx 8.73[\/latex] centimeters<\/p>\n

53.\u00a0[latex]\\frac{21\\pi }{10}\\approx 6.60[\/latex] meters<\/p>\n

55.\u00a0104.7198 cm2<\/p>\n

57.\u00a00.7697 in2<\/p>\n

59.\u00a0250\u00b0<\/p>\n

61.\u00a0320\u00b0<\/p>\n

63.\u00a0[latex]\\frac{4\\pi }{3}[\/latex]<\/p>\n

65.\u00a0[latex]\\frac{8\\pi }{9}[\/latex]<\/p>\n

67.\u00a01320 rad 210.085 RPM<\/p>\n

69.\u00a07 in.\/s, 4.77 RPM, 28.65 deg\/s<\/p>\n

71.\u00a0[latex]1,809,557.37\\text{ mm\/min}=30.16\\text{ m\/s}[\/latex]<\/p>\n

73.\u00a0[latex]5.76[\/latex]\u00a0miles<\/p>\n

75.\u00a0[latex]120^\\circ[\/latex]<\/p>\n

77.\u00a0794 miles per hour<\/p>\n

79.\u00a02,234 miles per hour<\/p>\n

81.\u00a011.5 inches<\/p>\n

Section 4.2 Solutions<\/h2>\n

1. The unit circle is a circle of radius 1 centered at the origin.<\/p>\n

3.\u00a0Yes, when the reference angle is [latex]\\frac{\\pi }{4}[\/latex] and the terminal side of the angle is in quadrants I and III. Thus, at [latex]x=\\frac{\\pi }{4},\\frac{5\\pi }{4}[\/latex], the sine and cosine values are equal.<\/p>\n

5.\u00a0Substitute the sine of the angle in for [latex]y[\/latex] in the Pythagorean Theorem [latex]{x}^{2}+{y}^{2}=1[\/latex]. Solve for [latex]x[\/latex] and take the negative solution.<\/p>\n

7. I<\/p>\n

9. IV<\/p>\n

11.\u00a0[latex]\\frac{\\sqrt{3}}{2}\\text{ , }\\frac{2\\sqrt{3}}{3}[\/latex]<\/p>\n

13.\u00a0[latex]\\frac{1}{2}\\text{ , }2[\/latex]<\/p>\n

15.\u00a0[latex]\\frac{\\sqrt{2}}{2}\\text{ , }\\sqrt{3}[\/latex]<\/p>\n

17. [latex]0{ , }\\sqrt{2}[\/latex]<\/p>\n

19. [latex]-1\\text{ , }0[\/latex]<\/p>\n

21. [latex]1\\text{ , }0[\/latex]<\/p>\n

23.\u00a0[latex]\\frac{\\sqrt{77}}{9}[\/latex]<\/p>\n

25.\u00a0[latex]-\\frac{\\sqrt{15}}{4}[\/latex]<\/p>\n

27.\u00a0[latex]\\sin t=\\frac{1}{2},\\csc t=2,\\cos t=-\\frac{\\sqrt{3}}{2},\\sec t=-\\frac{2\\sqrt{3}}{3},\\tan t=-\\frac{\\sqrt{3}}{3},\\cot t=-\\sqrt{3}[\/latex]<\/p>\n

29.\u00a0[latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\csc t=-\\sqrt{2},\\cos t=-\\frac{\\sqrt{2}}{2},\\sec t=-\\sqrt{2},\\tan t=1,\\cot t=1[\/latex]<\/p>\n

31.\u00a0[latex]\\sin t=\\frac{\\sqrt{3}}{2},\\csc t=\\frac{2\\sqrt{3}}{3},\\cos t=-\\frac{1}{2},\\sec t=-2,\\tan t=\\sqrt{3},\\cot t=\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\n

33.\u00a0[latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\csc t=-\\sqrt{2},\\cos t=\\frac{\\sqrt{2}}{2},\\sec t=\\sqrt{2},\\tan t=-1,\\cot t=1[\/latex]<\/p>\n

35.\u00a0[latex]\\sin t=0,\\csc t=\\varnothing,\\cos t=-1,\\sec t=-1,\\tan t=0,\\cot t=\\varnothing[\/latex]<\/p>\n

37.\u00a0[latex]\\sin t=-0.596,\\csc t=-1.679,\\cos t=0.803,\\sec t=1.245,\\tan t=-0.742,\\cot t=-1.347[\/latex]<\/p>\n

39.\u00a0\u22120.1736<\/p>\n

41.\u00a00.9511<\/p>\n

43.\u00a0\u22120.7071<\/p>\n

45.\u00a0\u22120.1392<\/p>\n

47.\u00a0\u22120.7660<\/p>\n

49.\u00a0\u20130.228<\/p>\n

51.\u00a0\u20132.414<\/p>\n

53.\u00a01.556<\/p>\n

55.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n

57.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n

59. 0<\/p>\n

61.\u00a0[latex]\\cos\\left(6t\\right)-\\sin\\left(9t\\right)[\/latex]<\/p>\n

63.\u00a0even<\/p>\n

65.\u00a0even<\/p>\n

67.\u00a013.77 hours, period: [latex]1000\\pi[\/latex]<\/p>\n

69.\u00a07.73 inches<\/p>\n

Section 4.3 Solutions<\/h2>\n

1.
\n\"A<\/p>\n

3. The tangent of an angle is the ratio of the opposite side to the adjacent side.<\/p>\n

5.\u00a0For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.<\/p>\n

7. [latex]\\frac{\\sqrt{2}-4}{4}[\/latex]<\/p>\n

9. 5<\/p>\n

11.\u00a0[latex]\\frac{\\pi }{6}[\/latex]<\/p>\n

13.\u00a0[latex]\\frac{\\pi }{4}[\/latex]<\/p>\n

15.\u00a0[latex]b=\\frac{20\\sqrt{3}}{3},c=\\frac{40\\sqrt{3}}{3}[\/latex]<\/p>\n

17.\u00a0[latex]a=10,000,c=10,000.5[\/latex]<\/p>\n

19.\u00a0[latex]b=\\frac{5\\sqrt{3}}{3},c=\\frac{10\\sqrt{3}}{3}[\/latex]<\/p>\n

21.\u00a0[latex]\\frac{5\\sqrt{29}}{29}[\/latex]<\/p>\n

23.\u00a0[latex]\\frac{5}{2}[\/latex]<\/p>\n

25.\u00a0[latex]\\frac{\\sqrt{29}}{2}[\/latex]<\/p>\n

27.\u00a0[latex]\\frac{5\\sqrt{41}}{41}[\/latex]<\/p>\n

29.\u00a0[latex]\\frac{5}{4}[\/latex]<\/p>\n

31.\u00a0[latex]\\frac{\\sqrt{41}}{4}[\/latex]<\/p>\n

33.\u00a0[latex]c=14, b=7\\sqrt{3}[\/latex]<\/p>\n

35.\u00a0[latex]a=15, b=15[\/latex]<\/p>\n

37.\u00a0[latex]b=9.9970, c=12.2041[\/latex]<\/p>\n

39.\u00a0[latex]a=2.0838, b=11.8177[\/latex]<\/p>\n

41.\u00a0[latex]a=55.9808,c=57.9555[\/latex]<\/p>\n

43.\u00a0[latex]a=46.6790,b=17.9184[\/latex]<\/p>\n

45.\u00a0[latex]a=16.4662,c=16.8341[\/latex]<\/p>\n

47.\u00a0188.3159<\/p>\n

49.\u00a0200.6737<\/p>\n

51.\u00a0498.3471 ft<\/p>\n

53.\u00a01060.09 ft<\/p>\n

55.\u00a027.372 ft<\/p>\n

57.\u00a022.6506 ft<\/p>\n

59.\u00a0368.7633 ft<\/p>\n

61. [latex]S 29.05^\\circ W[\/latex]<\/p>\n

63. East: 13.49 inches, North: 33.38 inches<\/p>\n

65. [latex]18.3^\\circ[\/latex]<\/p>\n

Section 4.4 Solutions<\/h2>\n

1.\u00a0Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, [latex]t[\/latex], formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis.<\/p>\n

3. The sine values are equal.<\/p>\n

5.\u00a0[latex]60^\\circ[\/latex]<\/p>\n

7.\u00a0[latex]80^\\circ[\/latex]<\/p>\n

9.\u00a0[latex]45^\\circ[\/latex]<\/p>\n

11.\u00a0[latex]\\frac{\\pi }{3}[\/latex]<\/p>\n

13.\u00a0[latex]\\frac{\\pi }{3}[\/latex]<\/p>\n

15.\u00a0[latex]\\frac{\\pi }{8}[\/latex]<\/p>\n

17.\u00a0[latex]60^\\circ[\/latex], Quadrant IV, [latex]\\text{sin}\\left(300^\\circ \\right)=-\\frac{\\sqrt{3}}{2},\\cos \\left(300^\\circ \\right)=\\frac{1}{2}[\/latex]<\/p>\n

19.\u00a0[latex]45^\\circ[\/latex], Quadrant II, [latex]\\text{sin}\\left(135^\\circ \\right)=\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\cos \\left(135^\\circ \\right)=-\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n

21.\u00a0[latex]60^\\circ[\/latex], Quadrant II, [latex]\\text{sin}\\left(480^\\circ \\right)=\\frac{\\sqrt{3}}{2}[\/latex], [latex]\\cos \\left(480^\\circ \\right)=-\\frac{1}{2}[\/latex]<\/p>\n

23.\u00a0[latex]30^\\circ[\/latex], Quadrant II, [latex]\\text{sin}\\left(-210^\\circ \\right)=\\frac{1}{2}[\/latex], [latex]\\cos \\left(-210^\\circ \\right)=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n

25.\u00a0[latex]\\frac{\\pi }{6}[\/latex], Quadrant III, [latex]\\text{sin}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{1}{2}[\/latex], [latex]\\text{cos}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n

27.\u00a0[latex]\\frac{\\pi }{4}[\/latex], Quadrant II, [latex]\\text{sin}\\left(\\frac{3\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\cos \\left(\\frac{4\\pi }{3}\\right)=-\\frac{\\sqrt[]{2}}{2}[\/latex]<\/p>\n

29.\u00a0[latex]\\frac{\\pi }{3}[\/latex], Quadrant II, [latex]\\text{sin}\\left(\\frac{2\\pi }{3}\\right)=\\frac{\\sqrt{3}}{2}[\/latex], [latex]\\cos \\left(\\frac{2\\pi }{3}\\right)=-\\frac{1}{2}[\/latex]<\/p>\n

31.\u00a0[latex]\\frac{\\pi }{4}[\/latex], Quadrant IV, [latex]\\text{sin}\\left(\\frac{-9\\pi }{4}\\right)=-\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\text{cos}\\left(\\frac{-9\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n

33.\u00a0[latex]\\frac{\\pi }{6}[\/latex], Quadrant III, [latex]\\text{sec}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{2\\sqrt{3}}{3}[\/latex]<\/p>\n

35.\u00a0[latex]\\frac{\\pi }{6}[\/latex], Quadrant I, [latex]\\text{cot}\\left(\\frac{13\\pi }{6}\\right)=\\sqrt{3}[\/latex]<\/p>\n

37.\u00a0[latex]\\frac{\\pi }{4}[\/latex], Quadrant II, [latex]\\text{sec}\\left(\\frac{3\\pi }{4}\\right)=-\\sqrt{2}[\/latex]<\/p>\n

39.\u00a0[latex]\\frac{\\pi }{4}[\/latex], Quadrant IV, [latex]\\text{cot}\\left(\\frac{11\\pi }{4}\\right)=-1[\/latex]<\/p>\n

41.\u00a0[latex]\\frac{\\pi }{3}[\/latex], Quadrant III, [latex]\\text{sec}\\left(-\\frac{2\\pi }{3}\\right)=-2[\/latex]<\/p>\n

43. [latex]\\frac{\\pi }{3}[\/latex], Quadrant IV, [latex]\\text{cot}\\left(-\\frac{7\\pi }{3}\\right)=-\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\n

45.\u00a0[latex]\\text{ }60^\\circ[\/latex], Quadrant IV, [latex]\\text{sec}\\left(300^\\circ\\right)=2[\/latex]<\/p>\n

47.\u00a0[latex]\\text{ }60^\\circ[\/latex], Quadrant III, [latex]\\text{cot}\\left(600^\\circ\\right)=\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\n

49.\u00a0[latex]\\text{ }30^\\circ[\/latex], Quadrant II, [latex]\\text{sec}\\left(-210^\\circ\\right)=-\\frac{2\\sqrt{3}}{3}[\/latex]<\/p>\n

51.\u00a0[latex]\\text{ }45^\\circ[\/latex], Quadrant IV, [latex]\\text{cot}\\left(-405^\\circ\\right)=-1[\/latex]<\/p>\n

53.\u00a0If [latex]\\text{ }\\sin t=-\\frac{2\\sqrt{2}}{3},\\sec t=-3,\\csc t=-\\frac{3\\sqrt{2}}{4},\\tan t=2\\sqrt{2},\\cot t=\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n

55.\u00a0[latex]\\text{ }\\sec t=2,\\csc t=\\frac{2\\sqrt{3}}{3},\\tan t=\\sqrt{3},\\cot t=\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\n

57.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n

59.\u00a0[latex]-\\frac{\\sqrt{6}}{4}[\/latex]<\/p>\n

61.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n

63.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n

65. 0<\/p>\n

Section 4.5 Solutions<\/h2>\n

1. The sine and cosine functions have the property that [latex]f(x+P)=f(x)[\/latex] for a certain P<\/em>. This means that the function values repeat for every P<\/em> units on the x-axis.<\/p>\n

3. The absolute value of the constant A<\/em> (amplitude) increases the total range and the constant D<\/em> (vertical shift) shifts the graph vertically.<\/p>\n

5. At the point where the terminal side of t<\/em> intersects the unit circle, you can determine that the sin t<\/em> equals the y<\/em>-coordinate of the point.<\/p>\n

7. amplitude: [latex]\\frac{2}{3}[\/latex]; period: 2\u03c0; midline: [latex]y=0[\/latex]; maximum: [latex]y=23[\/latex] occurs at [latex]x=0[\/latex]; minimum: [latex]y=\u221223[\/latex] occurs at [latex]x=\\pi[\/latex]; for one period, the graph starts at 0 and ends at 2\u03c0
\n\"A<\/p>\n

9. amplitude: 4; period: 2\u03c0; midline: [latex]y=0[\/latex]; maximum [latex]y=4[\/latex] occurs at [latex]x=\\frac{\\pi}{2}[\/latex]; minimum: [latex]y=\u22124[\/latex] occurs at [latex]x=\\frac{3\\pi}{2}[\/latex]; one full period occurs from [latex]x=0[\/latex] to [latex]x=2\u03c0[\/latex]
\n\"A<\/p>\n

11. amplitude: 1; period: \u03c0; midline: y=0; maximum: y=1 occurs at [latex]x=\\pi[\/latex]; minimum: [latex]y=\u22121[\/latex] occurs at [latex]x=\\frac{\\pi}{2}[\/latex]; one full period is graphed from [latex]x=0[\/latex] to [latex]x=\\pi[\/latex]
\n\"A<\/p>\n

13. amplitude: 4; period: 2; midline: [latex]y=0[\/latex]; maximum: [latex]y=4[\/latex] occurs at [latex]x=0[\/latex]; minimum: [latex]y=\u22124[\/latex] occurs at [latex]x=1[\/latex]
\n\"A<\/p>\n

15. amplitude: 3; period: [latex]\\frac{\\pi}{4}[\/latex]; midline: [latex]y=5[\/latex]; maximum: [latex]y=8[\/latex] occurs at [latex]x=0.12[\/latex]; minimum: [latex]y=2[\/latex] occurs at [latex]x=0.516[\/latex]; horizontal shift: \u22124; vertical translation 5; one period occurs from [latex]x=0[\/latex] to [latex]x=\\frac{\\pi}{4}[\/latex]
\n\"A<\/p>\n

17. amplitude: 5; period: [latex]\\frac{2\\pi}{5}; midline: [latex]y=\u22122[\/latex]; maximum: [latex]y=3[\/latex] occurs at [latex]x=0.08[\/latex]; minimum: [latex]y=\u22127[\/latex] occurs at [latex]x=0.71[\/latex]; phase shift:\u22124; vertical translation:\u22122; one full period can be graphed on [latex]x=0[\/latex] to [latex]x=\\frac{2\\pi}{5}[\/latex]
\n\"A<\/p>\n

19. amplitude: 1; period: 2\u03c0; midline: y=1; maximum:[latex]y=2[\/latex] occurs at [latex]x=2.09[\/latex]; maximum:[latex]y=2[\/latex] occurs at[latex]t=2.09[\/latex]; minimum:[latex]y=0[\/latex] occurs at [latex]t=5.24[\/latex]; phase shift: [latex]\u2212\\frac{\\pi}{3}[\/latex]; vertical translation: 1; one full period is from [latex]t=0[\/latex] to [latex]t=2\u03c0[\/latex]
\n\"A<\/p>\n

21. amplitude: 1; period: 4\u03c0; midline: [latex]y=0[\/latex]; maximum: [latex]y=1[\/latex] occurs at [latex]t=11.52[\/latex]; minimum: [latex]y=\u22121[\/latex] occurs at [latex]t=5.24[\/latex]; phase shift: \u2212[latex]\\frac{10\\pi}{3}[\/latex]; vertical shift: 0
\n\"A<\/p>\n

23. amplitude: 2; midline: [latex]y=\u22123[\/latex]; period: 4; equation: [latex]f(x)=2\\sin\\left(\\frac{\\pi}{2}x\\right)\u22123[\/latex]<\/p>\n

25. amplitude: 2; period: 5; midline: [latex]y=3[\/latex]; equation: [latex]f(x)=\u22122\\cos\\left(\\frac{2\\pi}{5}x\\right)+3[\/latex]<\/p>\n

27. amplitude: 4; period: 2; midline: [latex]y=0[\/latex]; equation: [latex]f(x)=\u22124\\cos\\left(\\pi\\left(x\u2212\\frac{\\pi}{2}\\right)\\right)[\/latex]<\/p>\n

29. amplitude: 2; period: 2; midline [latex]y=1[\/latex]; equation: [latex]f(x)=2\\cos\\left(\\frac{\\pi}{x}\\right)+1[\/latex]<\/p>\n

31.\u00a0[latex]\\frac{\\pi}{6},\\frac{5\\pi}{6}[\/latex]<\/p>\n

33.\u00a0[latex]\\frac{\\pi}{4},\\frac{3\\pi}{4}[\/latex]<\/p>\n

35.\u00a0[latex]\\frac{3\\pi}{2}[\/latex]<\/p>\n

37.\u00a0[latex]\\frac{\\pi}{2},\\frac{3\\pi}{2}[\/latex]<\/p>\n

39.\u00a0[latex]\\frac{\\pi}{2},\\frac{3\\pi}{2}[\/latex]<\/p>\n

41.\u00a0[latex]\\frac{\\pi}{6},\\frac{11\\pi}{6}[\/latex]<\/p>\n

43. The graph appears linear. The linear functions dominate the shape of the graph for large values of x.
\n\"A<\/p>\n

45. The graph is symmetric with respect to the y<\/em>-axis and there is no amplitude because the function is not periodic.
\n\"A<\/p>\n

47.
\na. Amplitude: 12.5; period: 10; midline: [latex]y=13.5[\/latex];
\nb. [latex]h(t)=12.5\\sin\\left(\\frac{\\pi}{5}\\left(t\u22122.5\\right)\\right)+13.5;[\/latex]
\nc. 26 ft<\/p>\n

Section 4.6 Solutions<\/h2>\n

1. \u00a0Since [latex]y=\\csc x[\/latex] is the reciprocal function of [latex]y=\\sin x[\/latex], you can plot the reciprocal of the coordinates on the graph of [latex]y=\\sin x[\/latex] to obtain the y<\/em>-coordinates of [latex]y=\\csc x[\/latex]. The x<\/em>-intercepts of the graph [latex]y=\\sin x[\/latex] are the vertical asymptotes for the graph of [latex]y=\\csc x[\/latex].<\/p>\n

3.\u00a0Answers will vary. Using the unit circle, one can show that [latex]\\tan(x+\\pi)=\\tan x[\/latex].<\/p>\n

5.\u00a0The period is the same: 2\u03c0.<\/p>\n

7. IV<\/p>\n

9. III<\/p>\n

11. period: 8; horizontal shift: 1 unit to left<\/p>\n

13. 1.5<\/p>\n

15. 5<\/p>\n

17. [latex]\u2212\\cot x\\cos x\u2212\\sin x[\/latex]<\/p>\n

19.\u00a0stretching factor: 2; period: [latex]\\frac{\\pi}{4}[\/latex]; asymptotes: [latex]x=\\frac{1}{4}\\left(\\frac{\\pi}{2}+\\pi k\\right)+8[\/latex], where k<\/em> is an integer
\n\"A<\/p>\n

21.\u00a0stretching factor: 6; period: 6; asymptotes: [latex]x=3k[\/latex], where k<\/em> is an integer
\n\"A<\/p>\n

23.\u00a0stretching factor: 1; period: \u03c0; asymptotes: [latex]x=\u03c0k[\/latex], where k<\/em> is an integer
\n\"A<\/p>\n

25.\u00a0Stretching factor: 1; period: \u03c0; asymptotes: [latex]x=\\frac{\\pi}{4}+{\\pi}k[\/latex], where k<\/em> is an integer
\n\"A<\/p>\n

27.\u00a0stretching factor: 2; period: 2\u03c0; asymptotes: [latex]x=\u03c0k[\/latex], where k<\/em> is an integer
\n\"A<\/p>\n

29.\u00a0stretching factor: 4; period: [latex]\\frac{2\\pi}{3}[\/latex]; asymptotes: [latex]x=\\frac{\\pi}{6}k[\/latex], where k<\/em> is an odd integer
\n\"A<\/p>\n

31.\u00a0stretching factor: 7; period: [latex]\\frac{2\\pi}{5}[\/latex]; asymptotes: [latex]x=\\frac{\\pi}{10}k[\/latex], where k<\/em> is an odd integer
\n\"A<\/p>\n

33.\u00a0stretching factor: 2; period: 2\u03c0; asymptotes: [latex]x=\u2212\\frac{\\pi}{4}+\\pi k[\/latex], where k<\/em> is an integer
\n\"A<\/p>\n

35.\u00a0stretching factor: [latex]\\frac{7}{5}[\/latex]; period: 2\u03c0; asymptotes: [latex]x=\\frac{\\pi}{4}+\\pi[\/latex]k<\/em>, where k<\/em> is an integer
\n\"A<\/p>\n

37. [latex]y=\\tan\\left(3\\left(x\u2212\\frac{\\pi}{4}\\right)\\right)+2[\/latex]
\n\"A<\/p>\n

39. [latex]f(x)=\\csc(2x)[\/latex]<\/p>\n

41. [latex]f(x)=\\csc(4x)[\/latex]<\/p>\n

43. [latex]f(x)=2\\csc x[\/latex]<\/p>\n

45. [latex]f(x)=\\frac{1}{2}\\tan(100\\pi x)[\/latex]<\/p>\n

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input [latex]\\csc x[\/latex] as [latex]\\frac{1}{\\sin x}[\/latex].<\/p>\n

46. [latex]f(x)=|\\csc(x)|[\/latex]<\/p>\n

47. [latex]f(x)=|\\cot(x)|[\/latex]<\/p>\n

48. [latex]f(x)=2^{\\csc(x)}[\/latex]<\/p>\n

49. [latex]f(x)=\\frac{\\csc(x)}{\\sec(x)}[\/latex]<\/p>\n

51.
\n\"A<\/p>\n

53.
\n\"A<\/p>\n

55. a. [latex](\u2212\\frac{\\pi}{2}\\text{,}\\frac{\\pi}{2})[\/latex];
\nb.
\n\"A
\nc. [latex]x=\u2212\\frac{\\pi}{2}[\/latex] and [latex]x=\\frac{\\pi}{2}[\/latex]; the distance grows without bound as |x<\/em>| approaches [latex]\\frac{\\pi}{2}[\/latex]\u2014i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;
\nd. 3; when [latex]x=\u2212\\frac{\\pi}{3}[\/latex], the boat is 3 km away;
\ne. 1.73; when [latex]x=\\frac{\\pi}{6}[\/latex], the boat is about 1.73 km away;
\nf. 1.5 km; when [latex]x=0[\/latex].<\/p>\n

57. a. [latex]h(x)=2\\tan\\left(\\frac{\\pi}{120}x\\right)[\/latex];
\nb.
\n\"An
\nc. [latex]h(0)=0:[\/latex] after 0 seconds, the rocket is 0 mi above the ground; [latex]h(30)=2:[\/latex] after 30 seconds, the rockets is 2 mi high;
\nd.\u00a0As x<\/em> approaches 60 seconds, the values of [latex]h(x)[\/latex] grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.<\/p>\n

Section 4.7 Solutions<\/h2>\n

1.\u00a0The function [latex]y=\\sin x[\/latex] is one-to-one on [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex]; thus, this interval is the range of the inverse function of [latex]y=\\sin x\\text{, }f\\left(x\\right)=\\sin^{\u22121}x[\/latex]. The function [latex]y=\\cos x[\/latex] is one-to-one on [0,\u03c0]; thus, this interval is the range of the inverse function of [latex]y=\\cos x\\text{, }f(x)=\\cos^{\u22121}x[\/latex].<\/p>\n

3. [latex]\\frac{\\pi}{6}[\/latex] is the radian measure of an angle between [latex]\u2212\\frac{\\pi}{2}[\/latex] and [latex]\\frac{\\pi}{2}[\/latex] whose sine is 0.5.<\/p>\n

5.\u00a0In order for any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex] so that it is one-to-one and possesses an inverse.<\/p>\n

7.\u00a0True. The angle, [latex]\\theta_{1}[\/latex] that equals [latex]\\arccos(\u2212x)\\text{, }x\\text{>}0[\/latex], will be a second quadrant angle with reference angle,\u00a0[latex]\\theta_{2}[\/latex], where [latex]\\theta_{2}[\/latex] equals [latex]\\arccos x\\text{, }x\\text{>}0[\/latex]. Since [latex]\\theta_{2}[\/latex] is the reference angle for [latex]\\theta_{1}[\/latex], [latex]\\theta_{2}=\\pi(\u2212x)=\\pi\u2212\\arccos x[\/latex]<\/p>\n

9. [latex]\u2212\\frac{\\pi}{6}[\/latex]<\/p>\n

11. [latex]\\frac{3\\pi}{4}[\/latex]<\/p>\n

13.\u00a0[latex]\u2212\\frac{\\pi}{3}[\/latex]<\/p>\n

15. [latex]\\frac{\\pi}{3}[\/latex]<\/p>\n

17. 1.98<\/p>\n

19. 0.93<\/p>\n

21. 1.41<\/p>\n

23. 0.56 radians<\/p>\n

25. 0<\/p>\n

27. 0.71<\/p>\n

29.\u00a0\u22120.71<\/p>\n

31. [latex]\u2212\\frac{\\pi}{4}[\/latex]<\/p>\n

33. 0.8<\/p>\n

35. [latex]\\frac{5}{13}[\/latex]<\/p>\n

37. [latex]\\frac{x}{\\sqrt{1\u2212x^{2}}}[\/latex]<\/p>\n

39. [latex]\\frac{\\sqrt{x^{2}\u22121}}{x}[\/latex]<\/p>\n

41. [latex]\\frac{2x}{\\sqrt{4x^{2}+1}}[\/latex]<\/p>\n

43. [latex]\\frac{\\sqrt{2x+1}}{x+1}[\/latex]<\/p>\n

45. [latex]\\frac{\\sqrt{2x+1}}{x}[\/latex]<\/p>\n

47.\u00a0[latex]\\frac{x}{\\sqrt{2x+1}}[\/latex]<\/p>\n

49.\u00a0domain [\u22121,1]; range [0,\u03c0]
\n\"A<\/p>\n

51. approximately [latex]x=0.00[\/latex]<\/p>\n

53. 0.395 radians<\/p>\n

55. 1.11 radians<\/p>\n

57. 1.25 radians<\/p>\n

59. 0.405 radians<\/p>\n

61. No. The angle the ladder makes with the horizontal is 60 degrees.<\/p>\n","protected":false},"author":264444,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-17057","chapter","type-chapter","status-publish","hentry"],"part":16602,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/17057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":28,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/17057\/revisions"}],"predecessor-version":[{"id":17639,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/17057\/revisions\/17639"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/parts\/16602"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/17057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/media?parent=17057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=17057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/contributor?post=17057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/license?post=17057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}