{"id":2349,"date":"2025-08-13T00:35:26","date_gmt":"2025-08-13T00:35:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2349"},"modified":"2026-02-17T23:14:46","modified_gmt":"2026-02-17T23:14:46","slug":"trigonometric-identities-and-equations","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/trigonometric-identities-and-equations\/","title":{"raw":"Trigonometric Identities and Equations: Get Stronger","rendered":"Trigonometric Identities and Equations: Get Stronger"},"content":{"raw":"

Simplifying Trigonometric Expressions With Identities<\/h1>\r\n3. After examining the reciprocal identity for [latex]\\sec t[\/latex], explain why the function is undefined at certain points.\r\n\r\nFor the following exercises, use the fundamental identities to fully simplify the expression.\r\n\r\n5. [latex]\\sin x\\cos x\\sec x[\/latex]\r\n\r\n7. [latex]\\tan x\\sin x+\\sec x{\\cos }^{2}x[\/latex]\r\n\r\n9.\u00a0[latex]\\frac{\\cot t+\\tan t}{\\sec \\left(-t\\right)}[\/latex]\r\n\r\n11. [latex]-\\tan \\left(-x\\right)\\cot \\left(-x\\right)[\/latex]\r\n\r\n13. [latex]\\frac{1+{\\tan }^{2}\\theta }{{\\csc }^{2}\\theta }+{\\sin }^{2}\\theta +\\frac{1}{{\\sec }^{2}\\theta }[\/latex]\r\n\r\n15. [latex]\\frac{1-{\\cos }^{2}x}{{\\tan }^{2}x}+2{\\sin }^{2}x[\/latex]\r\n\r\nFor the following exercises, verify the identity.\r\n\r\n29. [latex]\\cos x-{\\cos }^{3}x=\\cos x{\\sin }^{2}x[\/latex]\r\n\r\n31. [latex]\\frac{1+{\\sin }^{2}x}{{\\cos }^{2}x}=\\frac{1}{{\\cos }^{2}x}+\\frac{{\\sin }^{2}x}{{\\cos }^{2}x}=1+2{\\tan }^{2}x[\/latex]\r\n\r\n33. [latex]{\\cos }^{2}x-{\\tan }^{2}x=2-{\\sin }^{2}x-{\\sec }^{2}x[\/latex]\r\n\r\n39. [latex]\\frac{1+\\sin x}{\\cos x}=\\frac{\\cos x}{1+\\sin \\left(-x\\right)}[\/latex]\r\n

Sum and Difference Identities<\/h1>\r\n1. Explain the basis for the cofunction identities and when they apply.\r\n\r\nFor the following exercises, find the exact value.\r\n\r\n5. [latex]\\cos \\left(\\frac{\\pi }{12}\\right)[\/latex]\r\n\r\n7. [latex]\\sin \\left(\\frac{11\\pi }{12}\\right)[\/latex]\r\n\r\n9. [latex]\\tan \\left(\\frac{19\\pi }{12}\\right)[\/latex]\r\n\r\nFor the following exercises, rewrite in terms of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex].\r\n\r\n11. [latex]\\sin \\left(x-\\frac{3\\pi }{4}\\right)[\/latex]\r\n\r\n13. [latex]\\cos \\left(x+\\frac{2\\pi }{3}\\right)[\/latex]\r\n\r\nFor the following exercises, simplify the given expression.\r\n\r\n15. [latex]\\sec \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]\r\n\r\n17. [latex]\\tan \\left(\\frac{\\pi }{2}-x\\right)[\/latex]\r\n\r\n19. [latex]\\frac{\\tan \\left(\\frac{3}{2}x\\right)-\\tan \\left(\\frac{7}{5}x\\right)}{1+\\tan \\left(\\frac{3}{2}x\\right)\\tan \\left(\\frac{7}{5}x\\right)}[\/latex]\r\n\r\nFor the following exercise, find the requested information.\r\n\r\n21. Given that [latex]\\sin a=\\frac{4}{5}[\/latex], and [latex]\\cos b=\\frac{1}{3}[\/latex], with [latex]a[\/latex] and [latex]b[\/latex] both in the interval [latex]\\left[0,\\frac{\\pi }{2}\\right)[\/latex], find [latex]\\sin \\left(a-b\\right)[\/latex] and [latex]\\cos \\left(a+b\\right)[\/latex].\r\n\r\nFor the following exercise, find the exact value of each expression.\r\n\r\n23. [latex]\\cos \\left({\\cos }^{-1}\\left(\\frac{\\sqrt{2}}{2}\\right)+{\\sin }^{-1}\\left(\\frac{\\sqrt{3}}{2}\\right)\\right)[\/latex]\r\n\r\nFor the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical.\r\n\r\n25. [latex]\\cos \\left(\\frac{\\pi }{2}-x\\right)[\/latex]\r\n\r\n27. [latex]\\tan \\left(\\frac{\\pi }{3}+x\\right)[\/latex]\r\n\r\n31. [latex]\\sin \\left(\\frac{\\pi }{4}+x\\right)[\/latex]\r\n\r\nFor the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.\r\n\r\n43. [latex]\\sin \\left({195}^{\\circ }\\right)[\/latex]\r\n\r\n45. [latex]\\cos \\left({345}^{\\circ }\\right)[\/latex]\r\n\r\nFor the following exercises, prove the identities provided.\r\n\r\n47. [latex]\\tan \\left(x+\\frac{\\pi }{4}\\right)=\\frac{\\tan x+1}{1-\\tan x}[\/latex]\r\n\r\n49. [latex]\\frac{\\cos \\left(a+b\\right)}{\\cos a\\cos b}=1-\\tan a\\tan b[\/latex]\r\n\r\n51. [latex]\\frac{\\cos \\left(x+h\\right)-\\cos x}{h}=\\cos x\\frac{\\cos h - 1}{h}-\\sin x\\frac{\\sin h}{h}[\/latex]\r\n\r\nFor the following exercises, prove or disprove the statements.\r\n\r\n53. [latex]\\tan \\left(u-v\\right)=\\frac{\\tan u-\\tan v}{1+\\tan u\\tan v}[\/latex]\r\n\r\n55. If [latex]\\alpha ,\\beta [\/latex], and [latex]\\gamma [\/latex] are angles in the same triangle, then prove or disprove [latex]\\sin \\left(\\alpha +\\beta \\right)=\\sin \\gamma [\/latex].\r\n

Double Angle, Half Angle, and Reduction Formulas<\/h1>\r\n1. Explain how to determine the reduction identities from the double-angle identity [latex]\\cos \\left(2x\\right)={\\cos }^{2}x-{\\sin }^{2}x[\/latex].\r\n\r\nFor the following exercises, find the exact values of a) [latex]\\sin \\left(2x\\right)[\/latex], b) [latex]\\cos \\left(2x\\right)[\/latex], and c) [latex]\\tan \\left(2x\\right)[\/latex] without solving for [latex]x[\/latex].\r\n\r\n5. If [latex]\\sin x=\\frac{1}{8}[\/latex], and [latex]x[\/latex] is in quadrant I.\r\n\r\n7. If [latex]\\cos x=-\\frac{1}{2}[\/latex], and [latex]x[\/latex] is in quadrant III.\r\n\r\nFor the following exercises, find the exact value using half-angle formulas.\r\n\r\n15. [latex]\\sin \\left(\\frac{11\\pi }{12}\\right)[\/latex]\r\n\r\n17. [latex]\\tan \\left(\\frac{5\\pi }{12}\\right)[\/latex]\r\n\r\n19. [latex]\\tan \\left(-\\frac{3\\pi }{8}\\right)[\/latex]\r\n\r\nFor the following exercises, find the exact values of a) [latex]\\sin \\left(\\frac{x}{2}\\right)[\/latex], b) [latex]\\cos \\left(\\frac{x}{2}\\right)[\/latex], and c) [latex]\\tan \\left(\\frac{x}{2}\\right)[\/latex] without solving for [latex]x[\/latex].\r\n\r\n21. If [latex]\\sin x=-\\frac{12}{13}[\/latex], and [latex]x[\/latex] is in quadrant III.\r\n\r\n23. If [latex]\\sec x=-4[\/latex], and [latex]x[\/latex] is in quadrant II.\r\n\r\nFor the following exercises, use the triangle below to find the requested half and double angles.\r\n\r\n\"Image\r\n\r\n25. Find [latex]\\sin \\left(2\\alpha \\right),\\cos \\left(2\\alpha \\right)[\/latex], and [latex]\\tan \\left(2\\alpha \\right)[\/latex].\r\n\r\n27. Find [latex]\\sin \\left(\\frac{\\alpha }{2}\\right),\\cos \\left(\\frac{\\alpha }{2}\\right)[\/latex], and [latex]\\tan \\left(\\frac{\\alpha }{2}\\right)[\/latex].\r\n\r\nFor the following exercises, simplify each expression. Do not evaluate.\r\n\r\n29. [latex]2{\\cos }^{2}\\left({37}^{\\circ }\\right)-1[\/latex]\r\n\r\n31. [latex]{\\cos }^{2}\\left(9x\\right)-{\\sin }^{2}\\left(9x\\right)[\/latex]\r\n\r\n33. [latex]6\\sin \\left(5x\\right)\\cos \\left(5x\\right)[\/latex]\r\n\r\nFor the following exercises, prove the identity given.\r\n\r\n35. [latex]\\sin \\left(2x\\right)=-2\\sin \\left(-x\\right)\\cos \\left(-x\\right)[\/latex]\r\n\r\n37. [latex]\\frac{\\sin \\left(2\\theta \\right)}{1+\\cos \\left(2\\theta \\right)}{\\tan }^{2}\\theta =\\tan \\theta [\/latex]\r\n\r\nFor the following exercises, rewrite the expression with an exponent no higher than 1.\r\n\r\n39. [latex]{\\cos }^{2}\\left(6x\\right)[\/latex]\r\n\r\n41. [latex]{\\sin }^{4}\\left(3x\\right)[\/latex]\r\n\r\n43. [latex]{\\cos }^{4}x{\\sin }^{2}x[\/latex]\r\n\r\nFor the following exercises, reduce the equations to powers of one, and then check the answer graphically.\r\n\r\n45. [latex]{\\tan }^{4}x[\/latex]\r\n\r\n47. [latex]{\\sin }^{2}x{\\cos }^{2}x[\/latex]\r\n\r\n49. [latex]{\\tan }^{4}x{\\cos }^{2}x[\/latex]\r\n\r\n51. [latex]{\\cos }^{2}\\left(2x\\right)\\sin x[\/latex]\r\n\r\nFor the following exercises, prove the identities.\r\n\r\n55. [latex]\\sin \\left(2x\\right)=\\frac{2\\tan x}{1+{\\tan }^{2}x}[\/latex]\r\n\r\n57.\u00a0[latex]\\tan \\left(2x\\right)=\\frac{2\\sin x\\cos x}{2{\\cos }^{2}x - 1}[\/latex]\r\n\r\n59. [latex]\\sin \\left(3x\\right)=3\\sin x{\\cos }^{2}x-{\\sin }^{3}x[\/latex]\r\n\r\n61. [latex]\\frac{1+\\cos \\left(2t\\right)}{\\sin \\left(2t\\right)-\\cos t}=\\frac{2\\cos t}{2\\sin t - 1}[\/latex]\r\n\r\n63. [latex]\\cos \\left(16x\\right)=\\left({\\cos }^{2}\\left(4x\\right)-{\\sin }^{2}\\left(4x\\right)-\\sin \\left(8x\\right)\\right)\\left({\\cos }^{2}\\left(4x\\right)-{\\sin }^{2}\\left(4x\\right)+\\sin \\left(8x\\right)\\right)[\/latex]\r\n

Sum-to-Product and Product-to-Sum Formulas<\/h1>\r\nFor the following exercises, rewrite the product as a sum or difference.\r\n\r\n5. [latex]16\\sin \\left(16x\\right)\\sin \\left(11x\\right)[\/latex]\r\n\r\n7. [latex]2\\sin \\left(5x\\right)\\cos \\left(3x\\right)[\/latex]\r\n\r\n9. [latex]\\sin \\left(-x\\right)\\sin \\left(5x\\right)[\/latex]\r\n\r\nFor the following exercises, rewrite the sum or difference as a product.\r\n\r\n11. [latex]\\cos \\left(6t\\right)+\\cos \\left(4t\\right)[\/latex]\r\n\r\n13. [latex]\\cos \\left(7x\\right)+\\cos \\left(-7x\\right)[\/latex]\r\n\r\n15. [latex]\\cos \\left(3x\\right)+\\cos \\left(9x\\right)[\/latex]\r\n\r\nFor the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.\r\n\r\n17. [latex]\\cos \\left(45^\\circ \\right)\\cos \\left(15^\\circ \\right)[\/latex]\r\n\r\n19. [latex]\\sin \\left(-345^\\circ \\right)\\sin \\left(-15^\\circ \\right)[\/latex]\r\n\r\n21. [latex]\\sin \\left(-45^\\circ \\right)\\sin \\left(-15^\\circ \\right)[\/latex]\r\n\r\nFor the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.\r\n\r\n23. [latex]2\\sin \\left(100^\\circ \\right)\\sin \\left(20^\\circ \\right)[\/latex]\r\n\r\n25. [latex]\\sin \\left(213^\\circ \\right)\\cos \\left(8^\\circ \\right)[\/latex]\r\n\r\nFor the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.\r\n\r\n27. [latex]\\sin \\left(76^\\circ \\right)+\\sin \\left(14^\\circ \\right)[\/latex]\r\n\r\n29. [latex]\\sin \\left(101^\\circ \\right)-\\sin \\left(32^\\circ \\right)[\/latex]\r\n\r\n31. [latex]\\sin \\left(-1^\\circ \\right)+\\sin \\left(-2^\\circ \\right)[\/latex]\r\n\r\nFor the following exercises, prove the identity.\r\n\r\n33. [latex]4\\sin \\left(3x\\right)\\cos \\left(4x\\right)=2\\sin \\left(7x\\right)-2\\sin x[\/latex]\r\n\r\n35. [latex]\\sin x+\\sin \\left(3x\\right)=4\\sin x{\\cos }^{2}x[\/latex]\r\n\r\n37. [latex]2\\tan x\\cos \\left(3x\\right)=\\sec x\\left(\\sin \\left(4x\\right)-\\sin \\left(2x\\right)\\right)[\/latex]\r\n\r\nFor the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.\r\n\r\n39. [latex]\\cos \\left({58}^{\\circ }\\right)+\\cos \\left({12}^{\\circ }\\right)[\/latex]\r\n\r\n41. [latex]\\cos \\left({44}^{\\circ }\\right)-\\cos \\left({22}^{\\circ }\\right)[\/latex]\r\n\r\n43. [latex]\\sin \\left(-{14}^{\\circ }\\right)\\sin \\left({85}^{\\circ }\\right)[\/latex]\r\n\r\nFor the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.\r\n\r\n49. [latex]\\frac{\\sin \\left(9t\\right)-\\sin \\left(3t\\right)}{\\cos \\left(9t\\right)+\\cos \\left(3t\\right)}[\/latex]\r\n\r\n51. [latex]\\frac{\\sin \\left(3x\\right)-\\sin x}{\\sin x}[\/latex]\r\n\r\n53. [latex]\\sin x\\cos \\left(15x\\right)-\\cos x\\sin \\left(15x\\right)[\/latex]\r\n\r\nFor the following exercises, prove the identity.\r\n\r\n57. [latex]\\frac{\\cos \\left(3x\\right)+\\cos x}{\\cos \\left(3x\\right)-\\cos x}=-\\cot \\left(2x\\right)\\cot x[\/latex]\r\n\r\n59. [latex]\\frac{\\cos \\left(2y\\right)-\\cos \\left(4y\\right)}{\\sin \\left(2y\\right)+\\sin \\left(4y\\right)}=\\tan y[\/latex]\r\n\r\n61. [latex]\\cos x-\\cos \\left(3x\\right)=4{\\sin }^{2}x\\cos x[\/latex]\r\n\r\n63. [latex]\\tan \\left(\\frac{\\pi }{4}-t\\right)=\\frac{1-\\tan t}{1+\\tan t}[\/latex]\r\n

Solving Trigonometric Equations<\/h1>\r\n1. Will there always be solutions to trigonometric function equations? If not, describe an equation that would not have a solution. Explain why or why not.\r\n\r\n \r\n\r\nFor the following exercises, find all solutions exactly on the interval [latex]0\\le \\theta <2\\pi [\/latex].\r\n\r\n \r\n\r\n5. [latex]2\\sin \\theta =\\sqrt{3}[\/latex]\r\n\r\n \r\n\r\n7. [latex]2\\cos \\theta =-\\sqrt{2}[\/latex]\r\n\r\n \r\n\r\n9. [latex]\\tan x=1[\/latex]\r\n\r\n \r\n\r\n11. [latex]4{\\sin }^{2}x - 2=0[\/latex]\r\n\r\n \r\n\r\nFor the following exercises, solve exactly on [latex]\\left[0,2\\pi \\right)[\/latex].\r\n\r\n \r\n\r\n13. [latex]2\\cos \\theta =\\sqrt{2}[\/latex]\r\n\r\n \r\n\r\n15. [latex]2\\sin \\theta =-1[\/latex]\r\n\r\n \r\n\r\n17. [latex]2\\sin \\left(3\\theta \\right)=1[\/latex]\r\n\r\n \r\n\r\n19. [latex]2\\cos \\left(3\\theta \\right)=-\\sqrt{2}[\/latex]\r\n\r\n \r\n\r\n21. [latex]2\\sin \\left(\\pi \\theta \\right)=1[\/latex]\r\n\r\n \r\n\r\nFor the following exercises, factor to find all exact solutions on [latex]\\left[0,2\\pi \\right)[\/latex].\r\n\r\n \r\n\r\n23. [latex]\\sec \\left(x\\right)\\sin \\left(x\\right)-2\\sin \\left(x\\right)=0[\/latex]\r\n\r\n \r\n\r\n25. [latex]2{\\cos }^{2}t+\\cos \\left(t\\right)=1[\/latex]\r\n\r\n \r\n\r\n27. [latex]2\\sin \\left(x\\right)\\cos \\left(x\\right)-\\sin \\left(x\\right)+2\\cos \\left(x\\right)-1=0[\/latex]\r\n\r\n \r\n\r\n29. [latex]{\\sec }^{2}x=1[\/latex]\r\n\r\n \r\n\r\n31. [latex]8{\\sin }^{2}\\left(x\\right)+6\\sin \\left(x\\right)+1=0[\/latex]\r\n\r\n \r\n\r\nFor the following exercises, solve with the methods shown in this section exactly on the interval [latex]\\left[0,2\\pi \\right)[\/latex].\r\n\r\n \r\n\r\n33. [latex]\\sin \\left(3x\\right)\\cos \\left(6x\\right)-\\cos \\left(3x\\right)\\sin \\left(6x\\right)=-0.9[\/latex]\r\n\r\n \r\n\r\n35. [latex]\\cos \\left(2x\\right)\\cos x+\\sin \\left(2x\\right)\\sin x=1[\/latex]\r\n\r\n \r\n\r\n37. [latex]9\\cos \\left(2\\theta \\right)=9{\\cos }^{2}\\theta -4[\/latex]\r\n\r\n \r\n\r\n39. [latex]\\cos \\left(2t\\right)=\\sin t[\/latex]\r\n\r\n \r\n\r\nFor the following exercises, solve exactly on the interval [latex]\\left[0,2\\pi \\right)[\/latex]. Use the quadratic formula if the equations do not factor.\r\n\r\n \r\n\r\n41. [latex]{\\tan }^{2}x-\\sqrt{3}\\tan x=0[\/latex]\r\n\r\n \r\n\r\n43. [latex]{\\sin }^{2}x - 2\\sin x - 4=0[\/latex]\r\n\r\n \r\n\r\n45. [latex]3{\\cos }^{2}x - 2\\cos x - 2=0[\/latex]\r\n\r\n \r\n\r\n47. [latex]{\\tan }^{2}x+5\\tan x - 1=0[\/latex]\r\n\r\n \r\n\r\n49. [latex]-{\\tan }^{2}x-\\tan x - 2=0[\/latex]\r\n\r\n \r\n\r\nFor the following exercises, find exact solutions on the interval [latex]\\left[0,2\\pi \\right)[\/latex]. Look for opportunities to use trigonometric identities.\r\n\r\n \r\n\r\n51. [latex]{\\sin }^{2}x+{\\cos }^{2}x=0[\/latex]\r\n\r\n \r\n\r\n53. [latex]\\cos \\left(2x\\right)-\\cos x=0[\/latex]\r\n\r\n \r\n\r\n55. [latex]1-\\cos \\left(2x\\right)=1+\\cos \\left(2x\\right)[\/latex]\r\n\r\n \r\n\r\n57. [latex]10\\sin x\\cos x=6\\cos x[\/latex]\r\n\r\n \r\n\r\n59. [latex]4{\\cos }^{2}x - 4=15\\cos x[\/latex]\r\n\r\n \r\n\r\n61. [latex]8{\\cos }^{2}\\theta =3 - 2\\cos \\theta [\/latex]\r\n\r\n \r\n\r\n63. [latex]12{\\sin }^{2}t+\\cos t - 6=0[\/latex]\r\n\r\n \r\n\r\n65. [latex]{\\cos }^{3}t=\\cos t[\/latex]\r\n\r\nFor the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.\r\n\r\n \r\n\r\n67. [latex]8{\\cos }^{2}x - 2\\cos x - 1=0[\/latex]\r\n\r\n \r\n\r\n69. [latex]2{\\cos }^{2}x-\\cos x+15=0[\/latex]\r\n\r\n \r\n\r\n71. [latex]2{\\tan }^{2}x+7\\tan x+6=0[\/latex]\r\n\r\n \r\n\r\nFor the following exercises, use a calculator to find all solutions to four decimal places.\r\n\r\n \r\n\r\n73. [latex]\\sin x=0.27[\/latex]\r\n\r\n \r\n\r\n75. [latex]\\tan x=-0.34[\/latex]\r\n\r\n \r\n\r\nFor the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval [latex]\\left[0,2\\pi \\right)[\/latex]. Round to four decimal places.\r\n\r\n \r\n\r\n77. [latex]{\\tan }^{2}x+3\\tan x - 3=0[\/latex]\r\n\r\n \r\n\r\n79. [latex]{\\tan }^{2}x-\\sec x=1[\/latex]\r\n\r\n \r\n\r\n81. [latex]2{\\tan }^{2}x+9\\tan x - 6=0[\/latex]\r\n\r\n \r\n\r\nFor the following exercises, find all solutions exactly to the equations on the interval [latex]\\left[0,2\\pi \\right)[\/latex].\r\n\r\n \r\n\r\n83. [latex]{\\csc }^{2}x - 3\\csc x - 4=0[\/latex]\r\n\r\n \r\n\r\n85. [latex]{\\sin }^{2}x\\left(1-{\\sin }^{2}x\\right)+{\\cos }^{2}x\\left(1-{\\sin }^{2}x\\right)=0[\/latex]\r\n\r\n \r\n\r\n87. [latex]{\\sin }^{2}x - 1+2\\cos \\left(2x\\right)-{\\cos }^{2}x=1[\/latex]\r\n\r\n \r\n\r\n89. [latex]\\frac{\\sin \\left(2x\\right)}{{\\sec }^{2}x}=0[\/latex]\r\n\r\n \r\n\r\n91. [latex]2{\\cos }^{2}x-{\\sin }^{2}x-\\cos x - 5=0[\/latex]\r\n\r\n \r\n\r\n93. An airplane has only enough gas to fly to a city 200 miles northeast of its current location. If the pilot knows that the city is 25 miles north, how many degrees north of east should the airplane fly?\r\n\r\n \r\n\r\n95. If a loading ramp is placed next to a truck, at a height of 2 feet, and the ramp is 20 feet long, what angle does the ramp make with the ground?\r\n\r\n \r\n\r\n97. An astronaut is in a launched rocket currently 15 miles in altitude. If a man is standing 2 miles from the launch pad, at what angle is she looking down at him from horizontal? (Hint: this is called the angle of depression.)\r\n\r\n \r\n\r\n99. A man is standing 10 meters away from a 6-meter tall building. Someone at the top of the building is looking down at him. At what angle is the person looking at him?\r\n\r\n \r\n\r\n101. A 90-foot tall building has a shadow that is 2 feet long. What is the angle of elevation of the sun?\r\n\r\n \r\n\r\n103. A spotlight on the ground 3 feet from a 5-foot tall woman casts a 15-foot tall shadow on a wall 6 feet from the woman. At what angle is the light?\r\n\r\n \r\n\r\nFor the following exercises, find a solution to the following word problem algebraically. Then use a calculator to verify the result. Round the answer to the nearest tenth of a degree.\r\n\r\n \r\n\r\n105. A person does a handstand with her feet touching a wall and her hands 3 feet away from the wall. If the person is 5 feet tall, what angle do her feet make with the wall?\r\n

Modeling with Trigonometric Equations<\/span><\/h1>\r\n1. Explain what types of physical phenomena are best modeled by sinusoidal functions. What are the characteristics necessary?\r\n\r\n3. If we want to model cumulative rainfall over the course of a year, would a sinusoidal function be a good model? Why or why not?\r\n\r\nFor the following exercises, find a possible formula for the trigonometric function represented by the given table of values.\r\n\r\n5.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex] <\/strong><\/td>\r\n [latex]y[\/latex] <\/strong><\/td>\r\n<\/tr>\r\n
[latex]0[\/latex]<\/td>\r\n[latex]-4[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]3[\/latex]<\/td>\r\n[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]6[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]9[\/latex]<\/td>\r\n[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]12[\/latex]<\/td>\r\n[latex]-4[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]15[\/latex]<\/td>\r\n[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]18[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n7.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex] <\/strong><\/td>\r\n [latex]y[\/latex] <\/strong><\/td>\r\n<\/tr>\r\n
[latex]0[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\frac{\\pi }{4}[\/latex]<\/td>\r\n[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\frac{\\pi }{2}[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\frac{3\\pi }{4}[\/latex]<\/td>\r\n[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\pi [\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\frac{5\\pi }{4}[\/latex]<\/td>\r\n[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n9.<\/span>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex] <\/strong><\/td>\r\n [latex]y[\/latex] <\/strong><\/td>\r\n<\/tr>\r\n
[latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]1[\/latex]<\/td>\r\n[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]2[\/latex]<\/td>\r\n[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]3[\/latex]<\/td>\r\n[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]4[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]5[\/latex]<\/td>\r\n[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]6[\/latex]<\/td>\r\n[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n11.<\/span>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex] <\/strong><\/td>\r\n [latex]y[\/latex] <\/strong><\/td>\r\n<\/tr>\r\n
[latex]0[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]1[\/latex]<\/td>\r\n[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]2[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]3[\/latex]<\/td>\r\n[latex]13[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]4[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]5[\/latex]<\/td>\r\n[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]6[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n13.<\/span>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex] <\/strong><\/td>\r\n [latex]y[\/latex] <\/strong><\/td>\r\n<\/tr>\r\n
[latex]-1[\/latex]<\/td>\r\n[latex]\\sqrt{3}-2[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]0[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]1[\/latex]<\/td>\r\n[latex]2-\\sqrt{3}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]2[\/latex]<\/td>\r\n[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]3[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]4[\/latex]<\/td>\r\n[latex]\\sqrt{3}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]5[\/latex]<\/td>\r\n[latex]2+\\sqrt{3}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFor the following exercise, construct a function modeling behavior and use a calculator to find desired results.\r\n\r\n17. A city\u2019s average yearly rainfall is currently 20 inches and varies seasonally by 5 inches. Due to unforeseen circumstances, rainfall appears to be decreasing by 15% each year. How many years from now would we expect rainfall to initially reach 0 inches? Note, the model is invalid once it predicts negative rainfall, so choose the first point at which it goes below 0.\r\n\r\nFor the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.\r\n\r\n19. Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of [latex]84\\text{^\\circ F}[\/latex] occurs at 6PM and the average temperature for the day is [latex]70\\text{^\\circ F}\\text{.}[\/latex] Find the temperature, to the nearest degree, at 7AM.\r\n\r\n21. Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between [latex]64^\\circ\\text{F}[\/latex] and [latex]86^\\circ\\text{F}[\/latex] during the day and the average daily temperature first occurs at 12 AM. How many hours after midnight does the temperature first reach [latex]70^\\circ\\text{F?}[\/latex]\r\n\r\n23. A Ferris wheel is 45 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o\u2019clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. How many minutes of the ride are spent higher than 27 meters above the ground? Round to the nearest second\r\n\r\n25. The sea ice area around the South Pole fluctuates between about 18 million square kilometers in September to 3 million square kilometers in March. Assuming a sinusoidal fluctuation, when are there more than 15 million square kilometers of sea ice? Give your answer as a range of dates, to the nearest day.\r\n\r\n27. During a 90-day monsoon season, daily rainfall can be modeled by sinusoidal functions. A low of 4 inches of rainfall was recorded on day 30, and overall the average daily rainfall was 8 inches. During what period was daily rainfall less than 5 inches?\r\n\r\n29. In a certain region, monthly precipitation peaks at 24 inches in September and falls to a low of 4 inches in March. Identify the periods when the region is under flood conditions (greater than 22 inches) and drought conditions (less than 5 inches). Give your answer in terms of the nearest day.\r\n\r\nFor the following exercises, find the amplitude, period, and frequency of the given function.\r\n\r\n31. The displacement [latex]h\\left(t\\right)[\/latex] in centimeters of a mass suspended by a spring is modeled by the function [latex]h\\left(t\\right)=11\\sin \\left(12\\pi t\\right)[\/latex], where [latex]t[\/latex] is measured in seconds. Find the amplitude, period, and frequency of this displacement.\r\n\r\nFor the following exercises, construct an equation that models the described behavior.\r\n\r\n33. The displacement [latex]h\\left(t\\right)[\/latex], in centimeters, of a mass suspended by a spring is modeled by the function [latex]h\\left(t\\right)=-5\\cos \\left(60\\pi t\\right)[\/latex], where [latex]t[\/latex] is measured in seconds. Find the amplitude, period, and frequency of this displacement.\r\n\r\n35. A rabbit population oscillates 15 above and below average during the year, reaching the lowest value in January. The average population starts at 650 rabbits and increases by 110 each year. Find a function that models the population, [latex]P[\/latex], in terms of months since January, [latex]t[\/latex].\r\n\r\n37. A fish population oscillates 40 above and below average during the year, reaching the lowest value in January. The average population starts at 800 fish and increases by 4% each month. Find a function that models the population, [latex]P[\/latex], in terms of months since January, [latex]t[\/latex].\r\n\r\n39. A spring attached to the ceiling is pulled 7 cm down from equilibrium and released. The amplitude decreases by 11% each second. The spring oscillates 20 times each second. Find a function that models the distance, [latex]D[\/latex], the end of the spring is from equilibrium in terms of seconds, [latex]t[\/latex], since the spring was released.\r\n\r\n41. A spring attached to the ceiling is pulled 19 cm down from equilibrium and released. After 4 seconds, the amplitude has decreased to 14 cm. The spring oscillates 13 times each second. Find a function that models the distance, [latex]D[\/latex], the end of the spring is from equilibrium in terms of seconds, [latex]t[\/latex], since the spring was released.\r\n\r\nFor the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.\r\n\r\n43. Whitefish populations are currently at 500 in a lake. The population naturally oscillates above and below by 25 each year. If humans overfish, taking 4% of the population every year, in how many years will the lake first have fewer than 200 whitefish?\r\n\r\n45. A spring attached to a ceiling is pulled down 21 cm from equilibrium and released. After 6 seconds, the amplitude has decreased to 4 cm. The spring oscillates 20 times each second. Find when the spring first comes between [latex]-0.1[\/latex] and [latex]0.1\\text{ cm,}[\/latex] effectively at rest.\r\n\r\n47. Two springs are pulled down from the ceiling and released at the same time. The first spring, which oscillates 14 times per second, was initially pulled down 2 cm from equilibrium, and the amplitude decreases by 8% each second. The second spring, oscillating 22 times per second, was initially pulled down 10 cm from equilibrium and after 3 seconds has an amplitude of 2 cm. Which spring comes to rest first, and at what time? Consider \"rest\" as an amplitude less than [latex]0.1\\text{ cm}\\text{.}[\/latex]\r\n\r\n49. A plane flies 2 hours at 200 mph at a bearing of [latex]\\text{ }{60}^{\\circ }[\/latex], then continues to fly for 1.5 hours at the same speed, this time at a bearing of [latex]{150}^{\\circ }[\/latex]. Find the distance from the starting point and the bearing from the starting point. Hint: bearing is measured counterclockwise from north.","rendered":"

Simplifying Trigonometric Expressions With Identities<\/h1>\n

3. After examining the reciprocal identity for [latex]\\sec t[\/latex], explain why the function is undefined at certain points.<\/p>\n

For the following exercises, use the fundamental identities to fully simplify the expression.<\/p>\n

5. [latex]\\sin x\\cos x\\sec x[\/latex]<\/p>\n

7. [latex]\\tan x\\sin x+\\sec x{\\cos }^{2}x[\/latex]<\/p>\n

9.\u00a0[latex]\\frac{\\cot t+\\tan t}{\\sec \\left(-t\\right)}[\/latex]<\/p>\n

11. [latex]-\\tan \\left(-x\\right)\\cot \\left(-x\\right)[\/latex]<\/p>\n

13. [latex]\\frac{1+{\\tan }^{2}\\theta }{{\\csc }^{2}\\theta }+{\\sin }^{2}\\theta +\\frac{1}{{\\sec }^{2}\\theta }[\/latex]<\/p>\n

15. [latex]\\frac{1-{\\cos }^{2}x}{{\\tan }^{2}x}+2{\\sin }^{2}x[\/latex]<\/p>\n

For the following exercises, verify the identity.<\/p>\n

29. [latex]\\cos x-{\\cos }^{3}x=\\cos x{\\sin }^{2}x[\/latex]<\/p>\n

31. [latex]\\frac{1+{\\sin }^{2}x}{{\\cos }^{2}x}=\\frac{1}{{\\cos }^{2}x}+\\frac{{\\sin }^{2}x}{{\\cos }^{2}x}=1+2{\\tan }^{2}x[\/latex]<\/p>\n

33. [latex]{\\cos }^{2}x-{\\tan }^{2}x=2-{\\sin }^{2}x-{\\sec }^{2}x[\/latex]<\/p>\n

39. [latex]\\frac{1+\\sin x}{\\cos x}=\\frac{\\cos x}{1+\\sin \\left(-x\\right)}[\/latex]<\/p>\n

Sum and Difference Identities<\/h1>\n

1. Explain the basis for the cofunction identities and when they apply.<\/p>\n

For the following exercises, find the exact value.<\/p>\n

5. [latex]\\cos \\left(\\frac{\\pi }{12}\\right)[\/latex]<\/p>\n

7. [latex]\\sin \\left(\\frac{11\\pi }{12}\\right)[\/latex]<\/p>\n

9. [latex]\\tan \\left(\\frac{19\\pi }{12}\\right)[\/latex]<\/p>\n

For the following exercises, rewrite in terms of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex].<\/p>\n

11. [latex]\\sin \\left(x-\\frac{3\\pi }{4}\\right)[\/latex]<\/p>\n

13. [latex]\\cos \\left(x+\\frac{2\\pi }{3}\\right)[\/latex]<\/p>\n

For the following exercises, simplify the given expression.<\/p>\n

15. [latex]\\sec \\left(\\frac{\\pi }{2}-\\theta \\right)[\/latex]<\/p>\n

17. [latex]\\tan \\left(\\frac{\\pi }{2}-x\\right)[\/latex]<\/p>\n

19. [latex]\\frac{\\tan \\left(\\frac{3}{2}x\\right)-\\tan \\left(\\frac{7}{5}x\\right)}{1+\\tan \\left(\\frac{3}{2}x\\right)\\tan \\left(\\frac{7}{5}x\\right)}[\/latex]<\/p>\n

For the following exercise, find the requested information.<\/p>\n

21. Given that [latex]\\sin a=\\frac{4}{5}[\/latex], and [latex]\\cos b=\\frac{1}{3}[\/latex], with [latex]a[\/latex] and [latex]b[\/latex] both in the interval [latex]\\left[0,\\frac{\\pi }{2}\\right)[\/latex], find [latex]\\sin \\left(a-b\\right)[\/latex] and [latex]\\cos \\left(a+b\\right)[\/latex].<\/p>\n

For the following exercise, find the exact value of each expression.<\/p>\n

23. [latex]\\cos \\left({\\cos }^{-1}\\left(\\frac{\\sqrt{2}}{2}\\right)+{\\sin }^{-1}\\left(\\frac{\\sqrt{3}}{2}\\right)\\right)[\/latex]<\/p>\n

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical.<\/p>\n

25. [latex]\\cos \\left(\\frac{\\pi }{2}-x\\right)[\/latex]<\/p>\n

27. [latex]\\tan \\left(\\frac{\\pi }{3}+x\\right)[\/latex]<\/p>\n

31. [latex]\\sin \\left(\\frac{\\pi }{4}+x\\right)[\/latex]<\/p>\n

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.<\/p>\n

43. [latex]\\sin \\left({195}^{\\circ }\\right)[\/latex]<\/p>\n

45. [latex]\\cos \\left({345}^{\\circ }\\right)[\/latex]<\/p>\n

For the following exercises, prove the identities provided.<\/p>\n

47. [latex]\\tan \\left(x+\\frac{\\pi }{4}\\right)=\\frac{\\tan x+1}{1-\\tan x}[\/latex]<\/p>\n

49. [latex]\\frac{\\cos \\left(a+b\\right)}{\\cos a\\cos b}=1-\\tan a\\tan b[\/latex]<\/p>\n

51. [latex]\\frac{\\cos \\left(x+h\\right)-\\cos x}{h}=\\cos x\\frac{\\cos h - 1}{h}-\\sin x\\frac{\\sin h}{h}[\/latex]<\/p>\n

For the following exercises, prove or disprove the statements.<\/p>\n

53. [latex]\\tan \\left(u-v\\right)=\\frac{\\tan u-\\tan v}{1+\\tan u\\tan v}[\/latex]<\/p>\n

55. If [latex]\\alpha ,\\beta[\/latex], and [latex]\\gamma[\/latex] are angles in the same triangle, then prove or disprove [latex]\\sin \\left(\\alpha +\\beta \\right)=\\sin \\gamma[\/latex].<\/p>\n

Double Angle, Half Angle, and Reduction Formulas<\/h1>\n

1. Explain how to determine the reduction identities from the double-angle identity [latex]\\cos \\left(2x\\right)={\\cos }^{2}x-{\\sin }^{2}x[\/latex].<\/p>\n

For the following exercises, find the exact values of a) [latex]\\sin \\left(2x\\right)[\/latex], b) [latex]\\cos \\left(2x\\right)[\/latex], and c) [latex]\\tan \\left(2x\\right)[\/latex] without solving for [latex]x[\/latex].<\/p>\n

5. If [latex]\\sin x=\\frac{1}{8}[\/latex], and [latex]x[\/latex] is in quadrant I.<\/p>\n

7. If [latex]\\cos x=-\\frac{1}{2}[\/latex], and [latex]x[\/latex] is in quadrant III.<\/p>\n

For the following exercises, find the exact value using half-angle formulas.<\/p>\n

15. [latex]\\sin \\left(\\frac{11\\pi }{12}\\right)[\/latex]<\/p>\n

17. [latex]\\tan \\left(\\frac{5\\pi }{12}\\right)[\/latex]<\/p>\n

19. [latex]\\tan \\left(-\\frac{3\\pi }{8}\\right)[\/latex]<\/p>\n

For the following exercises, find the exact values of a) [latex]\\sin \\left(\\frac{x}{2}\\right)[\/latex], b) [latex]\\cos \\left(\\frac{x}{2}\\right)[\/latex], and c) [latex]\\tan \\left(\\frac{x}{2}\\right)[\/latex] without solving for [latex]x[\/latex].<\/p>\n

21. If [latex]\\sin x=-\\frac{12}{13}[\/latex], and [latex]x[\/latex] is in quadrant III.<\/p>\n

23. If [latex]\\sec x=-4[\/latex], and [latex]x[\/latex] is in quadrant II.<\/p>\n

For the following exercises, use the triangle below to find the requested half and double angles.<\/p>\n

\"Image<\/p>\n

25. Find [latex]\\sin \\left(2\\alpha \\right),\\cos \\left(2\\alpha \\right)[\/latex], and [latex]\\tan \\left(2\\alpha \\right)[\/latex].<\/p>\n

27. Find [latex]\\sin \\left(\\frac{\\alpha }{2}\\right),\\cos \\left(\\frac{\\alpha }{2}\\right)[\/latex], and [latex]\\tan \\left(\\frac{\\alpha }{2}\\right)[\/latex].<\/p>\n

For the following exercises, simplify each expression. Do not evaluate.<\/p>\n

29. [latex]2{\\cos }^{2}\\left({37}^{\\circ }\\right)-1[\/latex]<\/p>\n

31. [latex]{\\cos }^{2}\\left(9x\\right)-{\\sin }^{2}\\left(9x\\right)[\/latex]<\/p>\n

33. [latex]6\\sin \\left(5x\\right)\\cos \\left(5x\\right)[\/latex]<\/p>\n

For the following exercises, prove the identity given.<\/p>\n

35. [latex]\\sin \\left(2x\\right)=-2\\sin \\left(-x\\right)\\cos \\left(-x\\right)[\/latex]<\/p>\n

37. [latex]\\frac{\\sin \\left(2\\theta \\right)}{1+\\cos \\left(2\\theta \\right)}{\\tan }^{2}\\theta =\\tan \\theta[\/latex]<\/p>\n

For the following exercises, rewrite the expression with an exponent no higher than 1.<\/p>\n

39. [latex]{\\cos }^{2}\\left(6x\\right)[\/latex]<\/p>\n

41. [latex]{\\sin }^{4}\\left(3x\\right)[\/latex]<\/p>\n

43. [latex]{\\cos }^{4}x{\\sin }^{2}x[\/latex]<\/p>\n

For the following exercises, reduce the equations to powers of one, and then check the answer graphically.<\/p>\n

45. [latex]{\\tan }^{4}x[\/latex]<\/p>\n

47. [latex]{\\sin }^{2}x{\\cos }^{2}x[\/latex]<\/p>\n

49. [latex]{\\tan }^{4}x{\\cos }^{2}x[\/latex]<\/p>\n

51. [latex]{\\cos }^{2}\\left(2x\\right)\\sin x[\/latex]<\/p>\n

For the following exercises, prove the identities.<\/p>\n

55. [latex]\\sin \\left(2x\\right)=\\frac{2\\tan x}{1+{\\tan }^{2}x}[\/latex]<\/p>\n

57.\u00a0[latex]\\tan \\left(2x\\right)=\\frac{2\\sin x\\cos x}{2{\\cos }^{2}x - 1}[\/latex]<\/p>\n

59. [latex]\\sin \\left(3x\\right)=3\\sin x{\\cos }^{2}x-{\\sin }^{3}x[\/latex]<\/p>\n

61. [latex]\\frac{1+\\cos \\left(2t\\right)}{\\sin \\left(2t\\right)-\\cos t}=\\frac{2\\cos t}{2\\sin t - 1}[\/latex]<\/p>\n

63. [latex]\\cos \\left(16x\\right)=\\left({\\cos }^{2}\\left(4x\\right)-{\\sin }^{2}\\left(4x\\right)-\\sin \\left(8x\\right)\\right)\\left({\\cos }^{2}\\left(4x\\right)-{\\sin }^{2}\\left(4x\\right)+\\sin \\left(8x\\right)\\right)[\/latex]<\/p>\n

Sum-to-Product and Product-to-Sum Formulas<\/h1>\n

For the following exercises, rewrite the product as a sum or difference.<\/p>\n

5. [latex]16\\sin \\left(16x\\right)\\sin \\left(11x\\right)[\/latex]<\/p>\n

7. [latex]2\\sin \\left(5x\\right)\\cos \\left(3x\\right)[\/latex]<\/p>\n

9. [latex]\\sin \\left(-x\\right)\\sin \\left(5x\\right)[\/latex]<\/p>\n

For the following exercises, rewrite the sum or difference as a product.<\/p>\n

11. [latex]\\cos \\left(6t\\right)+\\cos \\left(4t\\right)[\/latex]<\/p>\n

13. [latex]\\cos \\left(7x\\right)+\\cos \\left(-7x\\right)[\/latex]<\/p>\n

15. [latex]\\cos \\left(3x\\right)+\\cos \\left(9x\\right)[\/latex]<\/p>\n

For the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.<\/p>\n

17. [latex]\\cos \\left(45^\\circ \\right)\\cos \\left(15^\\circ \\right)[\/latex]<\/p>\n

19. [latex]\\sin \\left(-345^\\circ \\right)\\sin \\left(-15^\\circ \\right)[\/latex]<\/p>\n

21. [latex]\\sin \\left(-45^\\circ \\right)\\sin \\left(-15^\\circ \\right)[\/latex]<\/p>\n

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.<\/p>\n

23. [latex]2\\sin \\left(100^\\circ \\right)\\sin \\left(20^\\circ \\right)[\/latex]<\/p>\n

25. [latex]\\sin \\left(213^\\circ \\right)\\cos \\left(8^\\circ \\right)[\/latex]<\/p>\n

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.<\/p>\n

27. [latex]\\sin \\left(76^\\circ \\right)+\\sin \\left(14^\\circ \\right)[\/latex]<\/p>\n

29. [latex]\\sin \\left(101^\\circ \\right)-\\sin \\left(32^\\circ \\right)[\/latex]<\/p>\n

31. [latex]\\sin \\left(-1^\\circ \\right)+\\sin \\left(-2^\\circ \\right)[\/latex]<\/p>\n

For the following exercises, prove the identity.<\/p>\n

33. [latex]4\\sin \\left(3x\\right)\\cos \\left(4x\\right)=2\\sin \\left(7x\\right)-2\\sin x[\/latex]<\/p>\n

35. [latex]\\sin x+\\sin \\left(3x\\right)=4\\sin x{\\cos }^{2}x[\/latex]<\/p>\n

37. [latex]2\\tan x\\cos \\left(3x\\right)=\\sec x\\left(\\sin \\left(4x\\right)-\\sin \\left(2x\\right)\\right)[\/latex]<\/p>\n

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.<\/p>\n

39. [latex]\\cos \\left({58}^{\\circ }\\right)+\\cos \\left({12}^{\\circ }\\right)[\/latex]<\/p>\n

41. [latex]\\cos \\left({44}^{\\circ }\\right)-\\cos \\left({22}^{\\circ }\\right)[\/latex]<\/p>\n

43. [latex]\\sin \\left(-{14}^{\\circ }\\right)\\sin \\left({85}^{\\circ }\\right)[\/latex]<\/p>\n

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.<\/p>\n

49. [latex]\\frac{\\sin \\left(9t\\right)-\\sin \\left(3t\\right)}{\\cos \\left(9t\\right)+\\cos \\left(3t\\right)}[\/latex]<\/p>\n

51. [latex]\\frac{\\sin \\left(3x\\right)-\\sin x}{\\sin x}[\/latex]<\/p>\n

53. [latex]\\sin x\\cos \\left(15x\\right)-\\cos x\\sin \\left(15x\\right)[\/latex]<\/p>\n

For the following exercises, prove the identity.<\/p>\n

57. [latex]\\frac{\\cos \\left(3x\\right)+\\cos x}{\\cos \\left(3x\\right)-\\cos x}=-\\cot \\left(2x\\right)\\cot x[\/latex]<\/p>\n

59. [latex]\\frac{\\cos \\left(2y\\right)-\\cos \\left(4y\\right)}{\\sin \\left(2y\\right)+\\sin \\left(4y\\right)}=\\tan y[\/latex]<\/p>\n

61. [latex]\\cos x-\\cos \\left(3x\\right)=4{\\sin }^{2}x\\cos x[\/latex]<\/p>\n

63. [latex]\\tan \\left(\\frac{\\pi }{4}-t\\right)=\\frac{1-\\tan t}{1+\\tan t}[\/latex]<\/p>\n

Solving Trigonometric Equations<\/h1>\n

1. Will there always be solutions to trigonometric function equations? If not, describe an equation that would not have a solution. Explain why or why not.<\/p>\n

 <\/p>\n

For the following exercises, find all solutions exactly on the interval [latex]0\\le \\theta <2\\pi[\/latex].\n\n \n\n5. [latex]2\\sin \\theta =\\sqrt{3}[\/latex]\n\n \n\n7. [latex]2\\cos \\theta =-\\sqrt{2}[\/latex]\n\n \n\n9. [latex]\\tan x=1[\/latex]\n\n \n\n11. [latex]4{\\sin }^{2}x - 2=0[\/latex]\n\n \n\nFor the following exercises, solve exactly on [latex]\\left[0,2\\pi \\right)[\/latex].\n\n \n\n13. [latex]2\\cos \\theta =\\sqrt{2}[\/latex]\n\n \n\n15. [latex]2\\sin \\theta =-1[\/latex]\n\n \n\n17. [latex]2\\sin \\left(3\\theta \\right)=1[\/latex]\n\n \n\n19. [latex]2\\cos \\left(3\\theta \\right)=-\\sqrt{2}[\/latex]\n\n \n\n21. [latex]2\\sin \\left(\\pi \\theta \\right)=1[\/latex]\n\n \n\nFor the following exercises, factor to find all exact solutions on [latex]\\left[0,2\\pi \\right)[\/latex].\n\n \n\n23. [latex]\\sec \\left(x\\right)\\sin \\left(x\\right)-2\\sin \\left(x\\right)=0[\/latex]\n\n \n\n25. [latex]2{\\cos }^{2}t+\\cos \\left(t\\right)=1[\/latex]\n\n \n\n27. [latex]2\\sin \\left(x\\right)\\cos \\left(x\\right)-\\sin \\left(x\\right)+2\\cos \\left(x\\right)-1=0[\/latex]\n\n \n\n29. [latex]{\\sec }^{2}x=1[\/latex]\n\n \n\n31. [latex]8{\\sin }^{2}\\left(x\\right)+6\\sin \\left(x\\right)+1=0[\/latex]\n\n \n\nFor the following exercises, solve with the methods shown in this section exactly on the interval [latex]\\left[0,2\\pi \\right)[\/latex].\n\n \n\n33. [latex]\\sin \\left(3x\\right)\\cos \\left(6x\\right)-\\cos \\left(3x\\right)\\sin \\left(6x\\right)=-0.9[\/latex]\n\n \n\n35. [latex]\\cos \\left(2x\\right)\\cos x+\\sin \\left(2x\\right)\\sin x=1[\/latex]\n\n \n\n37. [latex]9\\cos \\left(2\\theta \\right)=9{\\cos }^{2}\\theta -4[\/latex]\n\n \n\n39. [latex]\\cos \\left(2t\\right)=\\sin t[\/latex]\n\n \n\nFor the following exercises, solve exactly on the interval [latex]\\left[0,2\\pi \\right)[\/latex]. Use the quadratic formula if the equations do not factor.\n\n \n\n41. [latex]{\\tan }^{2}x-\\sqrt{3}\\tan x=0[\/latex]\n\n \n\n43. [latex]{\\sin }^{2}x - 2\\sin x - 4=0[\/latex]\n\n \n\n45. [latex]3{\\cos }^{2}x - 2\\cos x - 2=0[\/latex]\n\n \n\n47. [latex]{\\tan }^{2}x+5\\tan x - 1=0[\/latex]\n\n \n\n49. [latex]-{\\tan }^{2}x-\\tan x - 2=0[\/latex]\n\n \n\nFor the following exercises, find exact solutions on the interval [latex]\\left[0,2\\pi \\right)[\/latex]. Look for opportunities to use trigonometric identities.\n\n \n\n51. [latex]{\\sin }^{2}x+{\\cos }^{2}x=0[\/latex]\n\n \n\n53. [latex]\\cos \\left(2x\\right)-\\cos x=0[\/latex]\n\n \n\n55. [latex]1-\\cos \\left(2x\\right)=1+\\cos \\left(2x\\right)[\/latex]\n\n \n\n57. [latex]10\\sin x\\cos x=6\\cos x[\/latex]\n\n \n\n59. [latex]4{\\cos }^{2}x - 4=15\\cos x[\/latex]\n\n \n\n61. [latex]8{\\cos }^{2}\\theta =3 - 2\\cos \\theta[\/latex]\n\n \n\n63. [latex]12{\\sin }^{2}t+\\cos t - 6=0[\/latex]\n\n \n\n65. [latex]{\\cos }^{3}t=\\cos t[\/latex]\n\nFor the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.\n\n \n\n67. [latex]8{\\cos }^{2}x - 2\\cos x - 1=0[\/latex]\n\n \n\n69. [latex]2{\\cos }^{2}x-\\cos x+15=0[\/latex]\n\n \n\n71. [latex]2{\\tan }^{2}x+7\\tan x+6=0[\/latex]\n\n \n\nFor the following exercises, use a calculator to find all solutions to four decimal places.\n\n \n\n73. [latex]\\sin x=0.27[\/latex]\n\n \n\n75. [latex]\\tan x=-0.34[\/latex]\n\n \n\nFor the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval [latex]\\left[0,2\\pi \\right)[\/latex]. Round to four decimal places.\n\n \n\n77. [latex]{\\tan }^{2}x+3\\tan x - 3=0[\/latex]\n\n \n\n79. [latex]{\\tan }^{2}x-\\sec x=1[\/latex]\n\n \n\n81. [latex]2{\\tan }^{2}x+9\\tan x - 6=0[\/latex]\n\n \n\nFor the following exercises, find all solutions exactly to the equations on the interval [latex]\\left[0,2\\pi \\right)[\/latex].\n\n \n\n83. [latex]{\\csc }^{2}x - 3\\csc x - 4=0[\/latex]\n\n \n\n85. [latex]{\\sin }^{2}x\\left(1-{\\sin }^{2}x\\right)+{\\cos }^{2}x\\left(1-{\\sin }^{2}x\\right)=0[\/latex]\n\n \n\n87. [latex]{\\sin }^{2}x - 1+2\\cos \\left(2x\\right)-{\\cos }^{2}x=1[\/latex]\n\n \n\n89. [latex]\\frac{\\sin \\left(2x\\right)}{{\\sec }^{2}x}=0[\/latex]\n\n \n\n91. [latex]2{\\cos }^{2}x-{\\sin }^{2}x-\\cos x - 5=0[\/latex]\n\n \n\n93. An airplane has only enough gas to fly to a city 200 miles northeast of its current location. If the pilot knows that the city is 25 miles north, how many degrees north of east should the airplane fly?\n\n \n\n95. If a loading ramp is placed next to a truck, at a height of 2 feet, and the ramp is 20 feet long, what angle does the ramp make with the ground?\n\n \n\n97. An astronaut is in a launched rocket currently 15 miles in altitude. If a man is standing 2 miles from the launch pad, at what angle is she looking down at him from horizontal? (Hint: this is called the angle of depression.)\n\n \n\n99. A man is standing 10 meters away from a 6-meter tall building. Someone at the top of the building is looking down at him. At what angle is the person looking at him?\n\n \n\n101. A 90-foot tall building has a shadow that is 2 feet long. What is the angle of elevation of the sun?\n\n \n\n103. A spotlight on the ground 3 feet from a 5-foot tall woman casts a 15-foot tall shadow on a wall 6 feet from the woman. At what angle is the light?\n\n \n\nFor the following exercises, find a solution to the following word problem algebraically. Then use a calculator to verify the result. Round the answer to the nearest tenth of a degree.\n\n \n\n105. A person does a handstand with her feet touching a wall and her hands 3 feet away from the wall. If the person is 5 feet tall, what angle do her feet make with the wall?\n\n\n

Modeling with Trigonometric Equations<\/span><\/h1>\n

1. Explain what types of physical phenomena are best modeled by sinusoidal functions. What are the characteristics necessary?<\/p>\n

3. If we want to model cumulative rainfall over the course of a year, would a sinusoidal function be a good model? Why or why not?<\/p>\n

For the following exercises, find a possible formula for the trigonometric function represented by the given table of values.<\/p>\n

5.<\/p>\n\n\n\n\n\n\n\n\n\n\n
[latex]x[\/latex] <\/strong><\/td>\n [latex]y[\/latex] <\/strong><\/td>\n<\/tr>\n
[latex]0[\/latex]<\/td>\n[latex]-4[\/latex]<\/td>\n<\/tr>\n
[latex]3[\/latex]<\/td>\n[latex]-1[\/latex]<\/td>\n<\/tr>\n
[latex]6[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
[latex]9[\/latex]<\/td>\n[latex]-1[\/latex]<\/td>\n<\/tr>\n
[latex]12[\/latex]<\/td>\n[latex]-4[\/latex]<\/td>\n<\/tr>\n
[latex]15[\/latex]<\/td>\n[latex]-1[\/latex]<\/td>\n<\/tr>\n
[latex]18[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

7.<\/p>\n\n\n\n\n\n\n\n\n\n\n
[latex]x[\/latex] <\/strong><\/td>\n [latex]y[\/latex] <\/strong><\/td>\n<\/tr>\n
[latex]0[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
[latex]\\frac{\\pi }{4}[\/latex]<\/td>\n[latex]7[\/latex]<\/td>\n<\/tr>\n
[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
[latex]\\frac{3\\pi }{4}[\/latex]<\/td>\n[latex]-3[\/latex]<\/td>\n<\/tr>\n
[latex]\\pi[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
[latex]\\frac{5\\pi }{4}[\/latex]<\/td>\n[latex]7[\/latex]<\/td>\n<\/tr>\n
[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n


\n9.<\/span><\/p>\n\n\n\n\n\n\n\n\n\n\n
[latex]x[\/latex] <\/strong><\/td>\n [latex]y[\/latex] <\/strong><\/td>\n<\/tr>\n
[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n<\/tr>\n
[latex]1[\/latex]<\/td>\n[latex]-3[\/latex]<\/td>\n<\/tr>\n
[latex]2[\/latex]<\/td>\n[latex]-7[\/latex]<\/td>\n<\/tr>\n
[latex]3[\/latex]<\/td>\n[latex]-3[\/latex]<\/td>\n<\/tr>\n
[latex]4[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n<\/tr>\n
[latex]5[\/latex]<\/td>\n[latex]-3[\/latex]<\/td>\n<\/tr>\n
[latex]6[\/latex]<\/td>\n[latex]-7[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n


\n11.<\/span><\/p>\n\n\n\n\n\n\n\n\n\n\n
[latex]x[\/latex] <\/strong><\/td>\n [latex]y[\/latex] <\/strong><\/td>\n<\/tr>\n
[latex]0[\/latex]<\/td>\n[latex]5[\/latex]<\/td>\n<\/tr>\n
[latex]1[\/latex]<\/td>\n[latex]-3[\/latex]<\/td>\n<\/tr>\n
[latex]2[\/latex]<\/td>\n[latex]5[\/latex]<\/td>\n<\/tr>\n
[latex]3[\/latex]<\/td>\n[latex]13[\/latex]<\/td>\n<\/tr>\n
[latex]4[\/latex]<\/td>\n[latex]5[\/latex]<\/td>\n<\/tr>\n
[latex]5[\/latex]<\/td>\n[latex]-3[\/latex]<\/td>\n<\/tr>\n
[latex]6[\/latex]<\/td>\n[latex]5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n


\n13.<\/span><\/p>\n\n\n\n\n\n\n\n\n\n\n
[latex]x[\/latex] <\/strong><\/td>\n [latex]y[\/latex] <\/strong><\/td>\n<\/tr>\n
[latex]-1[\/latex]<\/td>\n[latex]\\sqrt{3}-2[\/latex]<\/td>\n<\/tr>\n
[latex]0[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n<\/tr>\n
[latex]1[\/latex]<\/td>\n[latex]2-\\sqrt{3}[\/latex]<\/td>\n<\/tr>\n
[latex]2[\/latex]<\/td>\n[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\n<\/tr>\n
[latex]3[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n<\/tr>\n
[latex]4[\/latex]<\/td>\n[latex]\\sqrt{3}[\/latex]<\/td>\n<\/tr>\n
[latex]5[\/latex]<\/td>\n[latex]2+\\sqrt{3}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

For the following exercise, construct a function modeling behavior and use a calculator to find desired results.<\/p>\n

17. A city\u2019s average yearly rainfall is currently 20 inches and varies seasonally by 5 inches. Due to unforeseen circumstances, rainfall appears to be decreasing by 15% each year. How many years from now would we expect rainfall to initially reach 0 inches? Note, the model is invalid once it predicts negative rainfall, so choose the first point at which it goes below 0.<\/p>\n

For the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.<\/p>\n

19. Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of [latex]84\\text{^\\circ F}[\/latex] occurs at 6PM and the average temperature for the day is [latex]70\\text{^\\circ F}\\text{.}[\/latex] Find the temperature, to the nearest degree, at 7AM.<\/p>\n

21. Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between [latex]64^\\circ\\text{F}[\/latex] and [latex]86^\\circ\\text{F}[\/latex] during the day and the average daily temperature first occurs at 12 AM. How many hours after midnight does the temperature first reach [latex]70^\\circ\\text{F?}[\/latex]<\/p>\n

23. A Ferris wheel is 45 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o\u2019clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. How many minutes of the ride are spent higher than 27 meters above the ground? Round to the nearest second<\/p>\n

25. The sea ice area around the South Pole fluctuates between about 18 million square kilometers in September to 3 million square kilometers in March. Assuming a sinusoidal fluctuation, when are there more than 15 million square kilometers of sea ice? Give your answer as a range of dates, to the nearest day.<\/p>\n

27. During a 90-day monsoon season, daily rainfall can be modeled by sinusoidal functions. A low of 4 inches of rainfall was recorded on day 30, and overall the average daily rainfall was 8 inches. During what period was daily rainfall less than 5 inches?<\/p>\n

29. In a certain region, monthly precipitation peaks at 24 inches in September and falls to a low of 4 inches in March. Identify the periods when the region is under flood conditions (greater than 22 inches) and drought conditions (less than 5 inches). Give your answer in terms of the nearest day.<\/p>\n

For the following exercises, find the amplitude, period, and frequency of the given function.<\/p>\n

31. The displacement [latex]h\\left(t\\right)[\/latex] in centimeters of a mass suspended by a spring is modeled by the function [latex]h\\left(t\\right)=11\\sin \\left(12\\pi t\\right)[\/latex], where [latex]t[\/latex] is measured in seconds. Find the amplitude, period, and frequency of this displacement.<\/p>\n

For the following exercises, construct an equation that models the described behavior.<\/p>\n

33. The displacement [latex]h\\left(t\\right)[\/latex], in centimeters, of a mass suspended by a spring is modeled by the function [latex]h\\left(t\\right)=-5\\cos \\left(60\\pi t\\right)[\/latex], where [latex]t[\/latex] is measured in seconds. Find the amplitude, period, and frequency of this displacement.<\/p>\n

35. A rabbit population oscillates 15 above and below average during the year, reaching the lowest value in January. The average population starts at 650 rabbits and increases by 110 each year. Find a function that models the population, [latex]P[\/latex], in terms of months since January, [latex]t[\/latex].<\/p>\n

37. A fish population oscillates 40 above and below average during the year, reaching the lowest value in January. The average population starts at 800 fish and increases by 4% each month. Find a function that models the population, [latex]P[\/latex], in terms of months since January, [latex]t[\/latex].<\/p>\n

39. A spring attached to the ceiling is pulled 7 cm down from equilibrium and released. The amplitude decreases by 11% each second. The spring oscillates 20 times each second. Find a function that models the distance, [latex]D[\/latex], the end of the spring is from equilibrium in terms of seconds, [latex]t[\/latex], since the spring was released.<\/p>\n

41. A spring attached to the ceiling is pulled 19 cm down from equilibrium and released. After 4 seconds, the amplitude has decreased to 14 cm. The spring oscillates 13 times each second. Find a function that models the distance, [latex]D[\/latex], the end of the spring is from equilibrium in terms of seconds, [latex]t[\/latex], since the spring was released.<\/p>\n

For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.<\/p>\n

43. Whitefish populations are currently at 500 in a lake. The population naturally oscillates above and below by 25 each year. If humans overfish, taking 4% of the population every year, in how many years will the lake first have fewer than 200 whitefish?<\/p>\n

45. A spring attached to a ceiling is pulled down 21 cm from equilibrium and released. After 6 seconds, the amplitude has decreased to 4 cm. The spring oscillates 20 times each second. Find when the spring first comes between [latex]-0.1[\/latex] and [latex]0.1\\text{ cm,}[\/latex] effectively at rest.<\/p>\n

47. Two springs are pulled down from the ceiling and released at the same time. The first spring, which oscillates 14 times per second, was initially pulled down 2 cm from equilibrium, and the amplitude decreases by 8% each second. The second spring, oscillating 22 times per second, was initially pulled down 10 cm from equilibrium and after 3 seconds has an amplitude of 2 cm. Which spring comes to rest first, and at what time? Consider “rest” as an amplitude less than [latex]0.1\\text{ cm}\\text{.}[\/latex]<\/p>\n

49. A plane flies 2 hours at 200 mph at a bearing of [latex]\\text{ }{60}^{\\circ }[\/latex], then continues to fly for 1.5 hours at the same speed, this time at a bearing of [latex]{150}^{\\circ }[\/latex]. Find the distance from the starting point and the bearing from the starting point. Hint: bearing is measured counterclockwise from north.<\/p>\n","protected":false},"author":67,"menu_order":40,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":201,"module-header":"practice","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2349"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2349\/revisions"}],"predecessor-version":[{"id":5708,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2349\/revisions\/5708"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/201"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2349\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2349"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2349"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2349"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2349"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}