"

Section 7.1 Answers

  1. [latex]30^{\circ}[/latex] is a Quadrant I angle
    coterminal with [latex]390^{\circ}[/latex] and [latex]-330^{\circ}[/latex]
    The rectangular plane with an angle, rotating in the counter-clockwise direction, with the terminal side in the first quadrant.
    Answer to Exercise 1
  2. [latex]120^{\circ}[/latex] is a Quadrant II angle
    coterminal with [latex]480^{\circ}[/latex] and [latex]-240^{\circ}[/latex]
    The rectangular plane with an angle, rotating in the counter-clockwise direction, with the terminal side in the second quadrant.
    Answer to Exercise 2
  3. [latex]225^{\circ}[/latex] is a Quadrant III angle
    coterminal with [latex]585^{\circ}[/latex] and [latex]-135^{\circ}[/latex]
    The rectangular plane with an angle, rotating in the counter-clockwise direction, with the terminal side in the third quadrant.
    Answer to Exercise 3
  4. [latex]330^{\circ}[/latex] is a Quadrant IV angle
    coterminal with [latex]690^{\circ}[/latex] and [latex]-30^{\circ}[/latex]
    The rectangular plane with an angle, rotating in the counter-clockwise direction, with the terminal side in the fourth quadrant.
    Answer to Exercise 4
  5. [latex]-30^{\circ}[/latex] is a Quadrant IV angle
    coterminal with [latex]330^{\circ}[/latex] and [latex]-390^{\circ}[/latex]
    The rectangular plane with an angle, rotating in the clockwise direction, with the terminal side in the fourth quadrant.
    Answer to Exercise 5
  6. [latex]-135^{\circ}[/latex] is a Quadrant III angle
    coterminal with [latex]225^{\circ}[/latex] and [latex]-495^{\circ}[/latex]
    The rectangular plane with an angle, rotating in the clockwise direction, with the terminal side in the third quadrant.
    Answer to Exercise 6
  7. [latex]-240^{\circ}[/latex] is a Quadrant II angle
    coterminal with [latex]120^{\circ}[/latex] and [latex]-600^{\circ}[/latex]
    The rectangular plane with an angle, rotating in the clockwise direction, with the terminal side in the second quadrant.
    Answer to Exercise 7
  8. [latex]-270^{\circ}[/latex] is a quadrantal angle
    coterminal with [latex]90^{\circ}[/latex] and [latex]-630^{\circ}[/latex]
    The rectangular plane with an angle, rotating in the clockwise direction, with the terminal side along the positive y-axis.
    Answer to Exercise 8
  9. [latex]405^{\circ}[/latex] is a Quadrant I angle
    coterminal with [latex]45^{\circ}[/latex] and [latex]-315^{\circ}[/latex]
    The rectangular plane with an angle, rotating in the counter-clockwise direction, with the terminal side in the first quadrant.
    Answer to Exercise 9
  10. [latex]840^{\circ}[/latex] is a Quadrant II angle
    coterminal with [latex]120^{\circ}[/latex] and [latex]-240^{\circ}[/latex]
    The rectangular plane with an angle, rotating in the counter-clockwise direction, with the terminal side in the second quadrant.
    Answer to Exercise 10
  11. [latex]-510^{\circ}[/latex] is a Quadrant III angle
    coterminal with [latex]-150^{\circ}[/latex] and [latex]210^{\circ}[/latex]
    The rectangular plane with an angle, rotating in the clockwise direction, with the terminal side in the third quadrant.
    Answer to Exercise 11
  12. [latex]-900^{\circ}[/latex] is a quadrantal angle
    coterminal with [latex]-180^{\circ}[/latex] and [latex]180^{\circ}[/latex]
    The rectangular plane with an angle, rotating in the clockwise direction, with the terminal side along the negative x-axis.
    Answer to Exercise 12
  13. [latex]\dfrac{\pi}{3}[/latex] is a Quadrant I angle
    coterminal with [latex]\dfrac{7\pi}{3}[/latex] and [latex]-\dfrac{5\pi}{3}[/latex]
    The rectangular plane with an angle, rotating in the counter-clockwise direction, with the terminal side in the first quadrant.
    Answer to Exercise 13
  14. [latex]\dfrac{5\pi}{6}[/latex] is a Quadrant II angle
    coterminal with [latex]\dfrac{17\pi}{6}[/latex] and [latex]-\dfrac{7\pi}{6}[/latex]
    The rectangular plane with an angle, rotating in the counter-clockwise direction, with the terminal side in the second quadrant.
    Answer to Exercise 14
  15. [latex]-\dfrac{11\pi}{3}[/latex] is a Quadrant I angle
    coterminal with [latex]\dfrac{\pi}{3}[/latex] and [latex]-\dfrac{5\pi}{3}[/latex]
    The rectangular plane with an angle, rotating the clockwise direction, with the terminal side in the first quadrant.
    Answer to Exercise 15
  16. [latex]\dfrac{5\pi}{4}[/latex] is a Quadrant III angle
    coterminal with [latex]\dfrac{13\pi}{4}[/latex] and [latex]-\dfrac{3\pi}{4}[/latex]
    The rectangular plane with an angle, rotating the counter-clockwise direction, with the terminal side in the third quadrant.
    Answer to Exercise 16
  17. [latex]\dfrac{3\pi}{4}[/latex] is a Quadrant II angle
    coterminal with [latex]\dfrac{11\pi}{4}[/latex] and [latex]-\dfrac{5\pi}{4}[/latex]
    The rectangular plane with an angle, rotating the counter-clockwise direction, with the terminal side in the second quadrant.
    Answer to Exercise 17
  18. [latex]-\dfrac{\pi}{3}[/latex] is a Quadrant IV angle
    coterminal with [latex]\dfrac{5\pi}{3}[/latex] and [latex]-\dfrac{7\pi}{3}[/latex]
    The rectangular plane with an angle, rotating the clockwise direction, with the terminal side in the fourth quadrant.
    Answer to Exercise 18
  19. [latex]\dfrac{7\pi}{2}[/latex] lies on the negative [latex]y[/latex]-axis
    coterminal with [latex]\dfrac{3\pi}{2}[/latex] and [latex]-\dfrac{\pi}{2}[/latex]
    The rectangular plane with an angle, rotating the counter-clockwise direction, with the terminal side along the negative y-axis.
    Answer to Exercise 19
  20. [latex]\dfrac{\pi}{4}[/latex] is a Quadrant I angle
    coterminal with [latex]\dfrac{9 \pi}{4}[/latex] and [latex]-\dfrac{7\pi}{4}[/latex]
    The rectangular plane with an angle, rotating the counter-clockwise direction, with the terminal side in the first quadrant.
    Answer to Exercise 20
  21. [latex]-\dfrac{\pi}{2}[/latex] lies on the negative [latex]y[/latex]-axis
    coterminal with [latex]\dfrac{3\pi}{2}[/latex] and [latex]-\dfrac{5\pi}{2}[/latex]
    The rectangular plane with an angle, rotating the clockwise direction, with the terminal side along the negative y-axis.
    Answer to Exercise 21
  22. [latex]\dfrac{7\pi}{6}[/latex] is a Quadrant III angle
    coterminal with [latex]\dfrac{19 \pi}{6}[/latex] and [latex]-\dfrac{5\pi}{6}[/latex]
    The rectangular plane with an angle, rotating the counter-clockwise direction, with the terminal side in the third quadrant.
    Answer to Exercise 22
  23. [latex]-\dfrac{5\pi}{3}[/latex] is a Quadrant I angle
    coterminal with [latex]\dfrac{\pi}{3}[/latex] and [latex]-\dfrac{11\pi}{3}[/latex]
    The rectangular plane with an angle, rotating the clockwise direction, with the terminal side in the first quadrant.
    Answer to Exercise 23
  24. [latex]3\pi[/latex] lies on the negative [latex]x[/latex]-axis
    coterminal with [latex]\pi[/latex] and [latex]-\pi[/latex]
    The rectangular plane with an angle, rotating the counter-clockwise direction, with the terminal side along the negative x-axis.
    Answer to Exercise 24
  25. [latex]-2\pi[/latex] lies on the positive [latex]x[/latex]-axis
    coterminal with [latex]2\pi[/latex] and [latex]-4\pi[/latex]
    The rectangular plane with an angle, rotating the clockwise direction, with the terminal side along the positive x-axis.
    Answer to Exercise 25
  26. [latex]-\dfrac{\pi}{4}[/latex] is a Quadrant IV angle
    coterminal with [latex]\dfrac{7 \pi}{4}[/latex] and [latex]-\dfrac{9\pi}{4}[/latex]
    The rectangular plane with an angle, rotating the clockwise direction, with the terminal side in the fourth quadrant.
    Answer to Exercise 26
  27. [latex]-\dfrac{\pi}{4}[/latex] is a Quadrant IV angle
    coterminal with [latex]\dfrac{7 \pi}{4}[/latex] and [latex]-\dfrac{9\pi}{4}[/latex]
    The rectangular plane with an angle, rotating the counter-clockwise direction, with the terminal side in the fourth quadrant.
    Answer to Exercise 27
  28. [latex]-\dfrac{13\pi}{6}[/latex] is a Quadrant IV angle
    coterminal with [latex]\dfrac{11\pi}{6}[/latex] and [latex]-\dfrac{\pi}{6}[/latex]
    The rectangular plane with an angle, rotating the clockwise direction, with the terminal side in the fourth quadrant.
    Answer to Exercise 28
  29. 0
  30. [latex]\dfrac{4\pi}{3}[/latex]
  31. [latex]\dfrac{3\pi}{4}[/latex]
  32. [latex]-\dfrac{3\pi}{2}[/latex]
  33. [latex]-\dfrac{7\pi}{4}[/latex]
  34. [latex]\dfrac{5\pi}{6}[/latex]
  35. [latex]\dfrac{\pi}{4}[/latex]
  36. [latex]-\dfrac{5\pi}{4}[/latex]
  37. [latex]180^{\circ}[/latex]
  38. [latex]-120^{\circ}[/latex]
  39. [latex]210^{\circ}[/latex]
  40. [latex]330^{\circ}[/latex]
  41. [latex]60^{\circ}[/latex]
  42. [latex]300^{\circ}[/latex]
  43. [latex]-30^{\circ}[/latex]
  44. [latex]90^{\circ}[/latex]
  45. [latex]t = \dfrac{5\pi}{6}[/latex]
    The unit circle with an arc, rotating in the counter-clockwise direction, that ends in the second quadrant.
    Answer to Exercise 45
  46. [latex]t = -\pi[/latex]
    The unit circle with an arc, rotating in the clockwise direction, that ends at the negative x-axis.
    Answer to Exercise 46
  47. [latex]t = 6[/latex]
    The unit circle with an arc, rotating in the counter-clockwise direction, that ends in fourth quadrant.
    Answer to Exercise 47
  48. [latex]t = -2[/latex]
    The unit circle with an arc, rotating in the clockwise direction, that ends in the third quadrant.
    Answer to Exercise 48
  49. [latex]t = 12[/latex] (between 1 and 2 revolutions)
    The unit circle with an arc, rotating in the counter-clockwise direction, that ends in fourth quadrant.
    Answer to Exercise 49
  50. [latex]\dfrac{3375 \pi}{352}[/latex] miles per hour
  51. [latex]\dfrac{19712}{\pi}[/latex] revolutions per minute
  52. [latex]\dfrac{35 \pi}{33}[/latex] miles per hour
  53. [latex]\dfrac{375 \pi}{22}[/latex] miles per hour
  54. 70 miles per hour
  55. [latex]\dfrac{1920 \pi}{1397}[/latex] miles per hour
  56. Answers May Vary
  57. [latex]12\pi[/latex] square units
  58. [latex]6250\pi[/latex] square units
  59. [latex]79.2825\pi \approx 249.07[/latex] square units
  60. [latex]\dfrac{\pi}{2}[/latex] square units
  61. [latex]\dfrac{50\pi}{3}[/latex] square units
  62. [latex]38.025 \pi \approx 119.46[/latex] square units

Section 7.2 Answers

  1. [latex]\theta = 30^{\circ}[/latex], [latex]a = 3\sqrt{3}[/latex], [latex]c = \sqrt{108} = 6\sqrt{3}[/latex]
  2. [latex]\alpha = 56^{\circ}[/latex], [latex]b = 12 \tan(34^{\circ}) = 8.094[/latex], [latex]c = 12\sec(34^{\circ}) = \dfrac{12}{\cos(34^{\circ})} \approx 14.475[/latex]
  3. [latex]\theta = 43^{\circ}[/latex], [latex]a = 6\cot(47^{\circ}) = \dfrac{6}{\tan(47^{\circ})} \approx 5.595[/latex], [latex]c = 6\csc(47^{\circ}) = \dfrac{6}{\sin(47^{\circ})} \approx 8.204[/latex]
  4. [latex]\beta = 40^{\circ}[/latex], [latex]b = 2.5 \tan(50^{\circ}) \approx 2.979[/latex], [latex]c = 2.5\sec(50^{\circ}) = \dfrac{2.5}{\cos(50^{\circ})} \approx 3.889[/latex]
  5. The side opposite [latex]\theta[/latex] has length [latex]10 \sin(15^{\circ}) \approx 2.588[/latex]
  6. The hypoteneuse has length [latex]14 \csc(38.2^{\circ}) = \dfrac{14}{\sin(38.2^{\circ})} \approx 22.639[/latex]
  7. The side adjacent to [latex]\theta[/latex] has length [latex]3.98 \cos(2.05^{\circ}) \approx 3.977[/latex]
  8. [latex]\cos(0) = 1[/latex], [latex]\; \sin(0) = 0[/latex]
  9. [latex]\cos \left(\dfrac{\pi}{4} \right) = \dfrac{\sqrt{2}}{2}[/latex], [latex]\; \sin \left(\dfrac{\pi}{4} \right) = \dfrac{\sqrt{2}}{2}[/latex]
  10. [latex]\cos \left(\dfrac{\pi}{3}\right) = \dfrac{1}{2}[/latex], [latex]\; \sin \left(\dfrac{\pi}{3}\right) = \dfrac{\sqrt{3}}{2}[/latex]
  11. [latex]\cos \left(\dfrac{\pi}{2}\right) = 0[/latex], [latex]\; \sin \left(\dfrac{\pi}{2}\right) = 1[/latex]
  12. [latex]\cos\left(\dfrac{2\pi}{3}\right) = -\dfrac{1}{2}[/latex], [latex]\; \sin \left(\dfrac{2\pi}{3}\right) = \dfrac{\sqrt{3}}{2}[/latex]
  13. [latex]\cos \left(\dfrac{3\pi}{4} \right) = -\dfrac{\sqrt{2}}{2}[/latex], [latex]\; \sin \left(\dfrac{3\pi}{4} \right) = \dfrac{\sqrt{2}}{2}[/latex]
  14. [latex]\cos(\pi) = -1[/latex], [latex]\; \sin(\pi) = 0[/latex]
  15. [latex]\cos\left(\dfrac{7\pi}{6}\right) = -\dfrac{\sqrt{3}}{2}[/latex], [latex]\; \sin\left(\dfrac{7\pi}{6}\right) = -\dfrac{1}{2}[/latex]
  16. [latex]\cos \left(\dfrac{5\pi}{4} \right) = -\dfrac{\sqrt{2}}{2}[/latex], [latex]\; \sin \left(\dfrac{5\pi}{4} \right) = -\dfrac{\sqrt{2}}{2}[/latex]
  17. [latex]\cos\left(\dfrac{4\pi}{3}\right) = -\dfrac{1}{2}[/latex], [latex]\; \sin \left(\dfrac{4\pi}{3}\right) = -\dfrac{\sqrt{3}}{2}[/latex]
  18. [latex]\cos \left(\dfrac{3\pi}{2}\right) = 0[/latex], [latex]\; \sin \left(\dfrac{3\pi}{2}\right) = -1[/latex]
  19. [latex]\cos\left(\dfrac{5\pi}{3}\right) = \dfrac{1}{2}[/latex], [latex]\; \sin \left(\dfrac{5\pi}{3}\right) = -\dfrac{\sqrt{3}}{2}[/latex]
  20. [latex]\cos \left(\dfrac{7\pi}{4} \right) = \dfrac{\sqrt{2}}{2}[/latex], [latex]\; \sin \left(\dfrac{7\pi}{4} \right) = -\dfrac{\sqrt{2}}{2}[/latex]
  21. [latex]\cos\left(\dfrac{23\pi}{6}\right) = \dfrac{\sqrt{3}}{2}[/latex], [latex]\; \sin\left(\dfrac{23\pi}{6}\right) = -\dfrac{1}{2}[/latex]
  22. [latex]\cos \left(-\dfrac{13\pi}{2}\right) = 0[/latex], [latex]\; \sin \left(-\dfrac{13\pi}{2}\right) = -1[/latex]
  23. [latex]\cos\left(-\dfrac{43\pi}{6}\right) = -\dfrac{\sqrt{3}}{2}[/latex], [latex]\; \sin\left(-\dfrac{43\pi}{6}\right) = \dfrac{1}{2}[/latex]
  24. [latex]\cos \left(-\dfrac{3\pi}{4} \right) = -\dfrac{\sqrt{2}}{2}[/latex], [latex]\; \sin \left(-\dfrac{3\pi}{4} \right) = -\dfrac{\sqrt{2}}{2}[/latex]
  25.  [latex]\cos\left(-\dfrac{\pi}{6}\right) = \dfrac{\sqrt{3}}{2}[/latex], [latex]\; \sin\left(-\dfrac{\pi}{6}\right) = -\dfrac{1}{2}[/latex]
  26. [latex]\cos\left(\dfrac{10\pi}{3}\right) = -\dfrac{1}{2}[/latex], [latex]\; \sin \left(\dfrac{10\pi}{3}\right) = -\dfrac{\sqrt{3}}{2}[/latex]
  27. [latex]\cos(117\pi) = -1[/latex], [latex]\; \sin(117\pi) = 0[/latex]
  28. [latex]\sin(\theta) = \dfrac{1}{2}[/latex] when [latex]\theta = \dfrac{\pi}{6} + 2\pi k[/latex] or [latex]\theta = \dfrac{5\pi}{6} + 2\pi k[/latex] for any integer [latex]k[/latex].
  29. [latex]\cos(\theta) = -\dfrac{\sqrt{3}}{2}[/latex] when [latex]\theta = \dfrac{5\pi}{6} + 2\pi k[/latex] or [latex]\theta = \dfrac{7\pi}{6} + 2\pi k[/latex] for any integer [latex]k[/latex].
  30. [latex]\sin(\theta) = 0[/latex] when [latex]\theta = \pi k[/latex] for any integer [latex]k[/latex].
  31. [latex]\cos(\theta) = \dfrac{\sqrt{2}}{2}[/latex] when [latex]\theta = \dfrac{\pi}{4} + 2\pi k[/latex] or [latex]\theta = \dfrac{7\pi}{4} + 2\pi k[/latex] for any integer [latex]k[/latex].
  32. [latex]\sin(\theta) = \dfrac{\sqrt{3}}{2}[/latex] when [latex]\theta = \dfrac{\pi}{3} + 2\pi k[/latex] or [latex]\theta = \dfrac{2\pi}{3} + 2\pi k[/latex] for any integer [latex]k[/latex].
  33. [latex]\cos(\theta) = -1[/latex] when [latex]\theta = (2k + 1)\pi[/latex] for any integer [latex]k[/latex].
  34. [latex]\sin(\theta) = -1[/latex] when [latex]\theta = \dfrac{3\pi}{2} + 2\pi k[/latex] for any integer [latex]k[/latex].
  35. [latex]\cos(\theta) = \dfrac{\sqrt{3}}{2}[/latex] when [latex]\theta = \dfrac{\pi}{6} + 2\pi k[/latex] or [latex]\theta = \dfrac{11\pi}{6} + 2\pi k[/latex] for any integer [latex]k[/latex].
  36. [latex]\cos(\theta) = -1.001[/latex] never happens
  37. [latex]\cos(t) = 0[/latex] when [latex]t = \dfrac{\pi}{2} + \pi k[/latex] for any integer [latex]k[/latex].
  38. [latex]\sin(t) = -\dfrac{\sqrt{2}}{2}[/latex] when [latex]t = \dfrac{5\pi}{4} + 2\pi k[/latex] or [latex]t = \dfrac{7\pi}{4} + 2\pi k[/latex] for any integer [latex]k[/latex].
  39. [latex]\cos(t) = 3[/latex] never happens.
  40. [latex]\sin(t) = -\dfrac{1}{2}[/latex] when [latex]t = \dfrac{7\pi}{6} + 2\pi k[/latex] or [latex]t = \dfrac{11\pi}{6} + 2\pi k[/latex] for any integer [latex]k[/latex].
  41. [latex]\cos(t) = \dfrac{1}{2}[/latex] when [latex]t = \dfrac{\pi}{3} + 2\pi k[/latex] or [latex]t = \dfrac{5\pi}{3} + 2\pi k[/latex] for any integer [latex]k[/latex].
  42. [latex]\sin(t) = -2[/latex] never happens
  43. [latex]\cos(t) = 1[/latex] when [latex]t = 2\pi k[/latex] for any integer [latex]k[/latex].
  44. [latex]\sin(t) = 1[/latex] when [latex]t = \dfrac{\pi}{2} + 2\pi k[/latex] for any integer [latex]k[/latex].
  45. [latex]\cos(t) = -\dfrac{\sqrt{2}}{2}[/latex] when [latex]t = \dfrac{3\pi}{4} + 2\pi k[/latex] or [latex]t = \dfrac{5\pi}{4} + 2\pi k[/latex] for any integer [latex]k[/latex].
  46. [latex]\cos(\theta) = -\dfrac{7}{25}, \; \sin(\theta) = \dfrac{24}{25}[/latex]
  47. [latex]\cos(\theta) = \dfrac{3}{5}, \; \sin(\theta) = \dfrac{4}{5}[/latex]
  48. [latex]\cos(\theta) = \dfrac{5\sqrt{106}}{106}, \; \sin(\theta) = -\dfrac{9\sqrt{106}}{106}[/latex]
  49. [latex]\cos(\theta) = -\dfrac{2\sqrt{5}}{25}, \; \sin(\theta) = -\dfrac{11\sqrt{5}}{25}[/latex]
  50. If [latex]\sin(\theta) = -\dfrac{7}{25}[/latex] with [latex]\theta[/latex] in Quadrant IV, then [latex]\cos(\theta) = \dfrac{24}{25}[/latex].
  51. If [latex]\cos(\theta) = \dfrac{4}{9}[/latex] with [latex]\theta[/latex] in Quadrant I, then [latex]\sin(\theta) = \dfrac{\sqrt{65}}{9}[/latex].
  52. If [latex]\sin(\theta) = \dfrac{5}{13}[/latex] with [latex]\theta[/latex] in Quadrant II, then [latex]\cos(\theta) = -\dfrac{12}{13}[/latex].
  53. If [latex]\cos(\theta) = -\dfrac{2}{11}[/latex] with [latex]\theta[/latex] in Quadrant III, then [latex]\sin(\theta) = -\dfrac{\sqrt{117}}{11}[/latex].
  54. If [latex]\sin(\theta) = -\dfrac{2}{3}[/latex] with [latex]\theta[/latex] in Quadrant III, then [latex]\cos(\theta) = -\dfrac{\sqrt{5}}{3}[/latex].
  55. If [latex]\cos(\theta) = \dfrac{28}{53}[/latex] with [latex]\theta[/latex] in Quadrant IV, then [latex]\sin(\theta) = -\dfrac{45}{53}[/latex].
  56. If [latex]\sin(\theta) = \dfrac{2\sqrt{5}}{5}[/latex] and [latex]\dfrac{\pi}{2}[/latex] < [latex]\theta[/latex] < [latex]\pi[/latex], then [latex]\cos(\theta) = -\dfrac{\sqrt{5}}{5}[/latex].
  57. If [latex]\cos(\theta) = \dfrac{\sqrt{10}}{10}[/latex] and [latex]2\pi[/latex] < [latex]\theta[/latex] < [latex]\dfrac{5\pi}{2}[/latex], then [latex]\sin(\theta) = \dfrac{3 \sqrt{10}}{10}[/latex].
  58. If [latex]\sin(\theta) = -0.42[/latex] and [latex]\pi[/latex] < [latex]\theta[/latex] < [latex]\dfrac{3\pi}{2}[/latex], then [latex]\cos(\theta) = -\sqrt{0.8236} \approx -0.9075[/latex].
  59. If [latex]\cos(\theta) = -0.98[/latex] and [latex]\dfrac{\pi}{2}[/latex] < [latex]\theta[/latex] < [latex]\pi[/latex], then [latex]\sin(\theta) = \sqrt{0.0396} \approx 0.1990[/latex].
  60. One solution is [latex]g(t) = 3t[/latex] and [latex]h(t) = \sin(2t)[/latex].
  61. One solution is [latex]g(\theta) = 3 \cos(\theta)[/latex] and [latex]h(\theta) = \sin(4 \theta)[/latex].
  62. One solution is [latex]g(t) = e^{-0.1t}[/latex] and [latex]h(t) = \sin(3t)[/latex].
  63. One solution is [latex]f(t) = \sin(t)[/latex] and [latex]g(t) = t[/latex].
  64. One solution is [latex]f(\theta) = 3 \cos(\theta)[/latex] and [latex]g(\theta) = \sqrt{\theta}[/latex].
  65. As we zoom in towards 0, the average rate of change of [latex]\sin(k t)[/latex] approaches [latex]k[/latex].
    A table with five rows and four columns.  The first row is the header row with S(t), [-0.1, 0.1], [-0.01, 0.01], and [-0.001, 0.001].  The first column has the functions sine of t, sine of 2t, sine of 3t, and sine of 4t.  The remaining entries are the values of the functions over the intervals.
    Answer to Exercise 65
  66. [latex]r = 1.125[/latex] inches, [latex]\omega = 9000 \pi \, \frac{\text{radians}}{\text{minute}}[/latex],  [latex]x = 1.125 \cos(9000 \pi \, t)[/latex], [latex]y = 1.125 \sin(9000 \pi \, t)[/latex].  Here [latex]x[/latex] and [latex]y[/latex] are measured in inches and [latex]t[/latex] is measured in minutes.
  67. [latex]r = 28[/latex] inches, [latex]\omega = \frac{2\pi}{3} \, \frac{\text{radians}}{\text{second}}[/latex],  [latex]x = 28 \cos\left(\frac{2\pi}{3} \, t \right)[/latex], [latex]y = 28 \sin\left(\frac{2\pi}{3} \, t \right)[/latex].  Here [latex]x[/latex] and [latex]y[/latex] are measured in inches and [latex]t[/latex] is measured in seconds.
  68. [latex]r = 1.25[/latex] inches, [latex]\omega = 14400 \pi \, \frac{\text{radians}}{\text{minute}}[/latex],  [latex]x = 1.25 \cos(14400 \pi \, t)[/latex], [latex]y = 1.25 \sin(14400 \pi \, t)[/latex].  Here [latex]x[/latex] and [latex]y[/latex] are measured in inches and [latex]t[/latex] is measured in minutes.
  69. [latex]r = 64[/latex] feet, [latex]\omega = \frac{4\pi}{127} \, \frac{\text{radians}}{\text{second}}[/latex],  [latex]x = 64 \cos\left(\frac{4\pi}{127} \, t \right)[/latex], [latex]y = 64 \sin\left(\frac{4\pi}{127} \, t \right)[/latex].  Here [latex]x[/latex] and [latex]y[/latex] are measured in feet and [latex]t[/latex] is

Section 7.3 Answers

  1. [latex]f(t) = 3\sin(t)[/latex]
    Period: [latex]2\pi[/latex]
    Amplitude: 3
    Phase Shift: 0
    Vertical Shift: 0
    A sine wave that starts at (0,0) and ends at (2 pi, 0) with amplitude of 3.
    Answer to Exercise 1
  2. [latex]g(t) = \sin(3t)[/latex]
    Period: [latex]\frac{2\pi}{3}[/latex]
    Amplitude: 1
    Phase Shift: 0
    Vertical Shift: 0
    A sine wave that starts at (0,0) and ends at (2 pi over 3, 0) with amplitude of 1.
    Answer to Exercise 2
  3. [latex]h(t) = -2\cos(t)[/latex]
    Period: [latex]2\pi[/latex]
    Amplitude: 2
    Phase Shift: 0
    Vertical Shift: 0
    A cosine wave that starts at (0,-2) and ends at (2 pi, -2) with an amplitude of 2.
    Answer to Exercise 3
  4. [latex]f(t) = \cos \left( t - \frac{\pi}{2} \right)[/latex]
    Period: [latex]2\pi[/latex]
    Amplitude: 1
    Phase Shift: [latex]\frac{\pi}{2}[/latex]
    Vertical Shift: 0
    A cosine wave that starts at (pi over 2, 1) and ends at (5 pi over 2, 1) with an amplitude of 3.
    Answer to Exercise 4
  5. [latex]g(t) = -\sin \left( t + \frac{\pi}{3} \right)[/latex]
    Period: [latex]2\pi[/latex]
    Amplitude: 1
    Phase Shift: [latex]-\frac{\pi}{3}[/latex]
    Vertical Shift: 0
    A sine wave that starts at (- pi over 3, 0) and ends at (5 pi over 3, 0) with an amplitude of 1.
    Answer to Exercise 5
  6. [latex]h(t) = \sin(2t - \pi)[/latex]
    Period: [latex]\pi[/latex]
    Amplitude: 1
    Phase Shift: [latex]\frac{\pi}{2}[/latex]
    Vertical Shift: 0
    A sine wave that starts at (pi over 2, 0) and ends at (3 pi over 2, 0) with an amplitude of 1.
    Answer to Exercise 6
  7. [latex]f(t) = -\frac{1}{3}\cos \left( \frac{1}{2}t + \frac{\pi}{3} \right)[/latex]
    Period: [latex]4\pi[/latex]
    Amplitude: [latex]\frac{1}{3}[/latex]
    Phase Shift: [latex]-\frac{2\pi}{3}[/latex]
    Vertical Shift: 0
    A cosine wave that starts at (-2 pi over 3, -1/3) and ends at (10 pi over 3, -1/3) with an amplitude of one third.
    Answer to Exercise 7
  8. [latex]g(t) = \cos (3t - 2\pi) + 4[/latex]
    Period: [latex]\frac{2\pi}{3}[/latex]
    Amplitude: 1
    Phase Shift: [latex]\frac{2\pi}{3}[/latex]
    Vertical Shift: 4
    A cosine wave that starts at (2 pi over 3, 5) and ends at (4 pi over 3, 5) with an amplitude of 1.
    Answer to Exercise 8
  9. [latex]h(t) = \sin \left( -t - \frac{\pi}{4} \right) - 2[/latex]
    Period: [latex]2\pi[/latex]
    Amplitude: 1
    Phase Shift: [latex]-\frac{\pi}{4}[/latex] (You need to use [latex]y = -\sin \left( t + \frac{\pi}{4} \right) - 2[/latex] to find this.)[1]
    Vertical Shift: [latex]-2[/latex]
    A sine wave over two cycles, starts at (-7 pi over 4, -2) and ends at (7 pi over 4, -2) with an amplitude of 1.
    Answer to Exercise 9
  10. [latex]f(t) = \frac{2}{3} \cos \left( \frac{\pi}{2} - 4t \right) + 1[/latex]
    Period: [latex]\frac{\pi}{2}[/latex]
    Amplitude: [latex]\frac{2}{3}[/latex]
    Phase Shift: [latex]\frac{\pi}{8}[/latex] (You need to use [latex]y = \frac{2}{3} \cos \left( 4t - \frac{\pi}{2} \right) + 1[/latex] to find this.)[2]
    Vertical Shift: 1
    A cosine function over two cycles, starts at (-3 pi over 8, 5/3) and ends at (5 pi over 8, 5/3) with an amplitude of 2/3.
    Answer to Exercise 10
  11. [latex]g(t) = -\frac{3}{2} \cos \left( 2t + \frac{\pi}{3} \right) - \frac{1}{2}[/latex]
    Period: [latex]\pi[/latex]
    Amplitude: [latex]\frac{3}{2}[/latex]
    Phase Shift: [latex]-\frac{\pi}{6}[/latex]
    Vertical Shift: [latex]-\frac{1}{2}[/latex]
    A cosine wave that starts (- pi over 6, -2) and ends at (5 pi over 6, -2) with an amplitude of 1.5.
    Answer to Exercise 11
  12. [latex]h(t) = 4\sin (-2\pi t + \pi)[/latex]
    Period: 1
    Amplitude: 4
    Phase Shift: [latex]\frac{1}{2}[/latex] (You need to use [latex]h(t) = -4\sin (2\pi t - \pi)[/latex] to find this.)[3]
    Vertical Shift: 0
    A sine wave that starts at (-0.5, 0) and ends at (1.5,0) with an amplitude of 4.
    Answer to Exercise 12
  13. [latex]S(t) = 4 \sin \left(t + \frac{\pi}{4} \right)[/latex], [latex]C(t) = 4 \cos \left(t - \frac{\pi}{4} \right)[/latex]
  14. [latex]S(t) = -3 \sin(t) + 3[/latex], [latex]C(t) = -3 \cos\left(t - \frac{\pi}{2}\right) + 3[/latex]
  15. [latex]S(t) = 3 \sin \left( 2t - \frac{\pi}{3} \right)[/latex], [latex]C(t) = 3 \cos \left( 2t - \frac{5\pi}{6} \right)[/latex]
  16. [latex]S(t) = \frac{7}{2} \sin(\pi t) + \frac{1}{2}[/latex], [latex]C(t) = \frac{7}{2} \cos\left(\pi t \frac{\pi}{2} \right) + \frac{1}{2}[/latex]
  17. .
    1. [latex]y = |4 \sin(t)|[/latex].  Period: [latex]\pi[/latex].

      Two cycles are graphed below.

      A curve that starts at (0,0) increases to (pi over 2, 4), decreases to (pi,0), increases to (3 pi over 2, 4) and ends at (2 pi, 0).
      Answer to Exercise 17a
    2. [latex]y = \sqrt{4 \sin(t)}[/latex].  Period: [latex]2\pi[/latex].

      One cycle is graphed below.

      A curve that starts at (0,0) and increases to (pi over 2, 2) and back to (pi,0).
      Answer to Exercise 17b
  18. [latex]f(t)=\cos(3t) + \sin(t)[/latex] over [latex][-2\pi, 2\pi][/latex]
    A sinusoidal wave.
    Answer to Exercise 18
  19. [latex]f(t)=\frac{\sin(t)}{t}[/latex] over [latex][-2\pi, 2\pi][/latex]
    A sinusoidal wave that goes through (0,1).
    Answer to Exercise 19
  20. [latex]f(t)=t\sin(t)[/latex] over [latex][-4\pi, 4\pi][/latex]
    A sinusoidal wave with the line y=x in blue and y=-x in red.  The wave period increases and the amplitude decreases as you get closer and closer to 0.
    Answer to Exercise 20
  21. [latex]f(t)=\sin\left(\frac{1}{t}\right)[/latex] over [latex][-\pi, \pi][/latex]
    A sinusoidal wave where wave period increases as you get closer and closer to 0.
    Answer to Exercise 21
  22. [latex]f(t)=e^{-0.1t}(\cos(2t) + \sin(2t)))[/latex] over [latex][-\pi, 3\pi][/latex]
    A sinusoidal wave whose amplitude decreases over the interval.  The graph includes the exponential function y=e raised to the -0.1t in blue and y = -1 times e raised to the -0.1t in red.
    Answer to Exercise 22
  23. [latex]f(t)=e^{-0.1t}(\cos(2t) + 2\sin(t)))[/latex] over [latex][-\pi, 3\pi][/latex]
    A sinusoidal wave.  The graph includes the exponential function y=e raised to the -0.1t in blue and y = -1 times e raised to the -0.1t in red.
    Answer to Exercise 23
  24. Answers May Vary
  25. [latex]S(t) = \sin\left(880\pi t\right)[/latex]
  26. [latex]V(t) = 220 \sqrt{2} \sin\left(120\pi t\right)[/latex]
  27. [latex]h(t) = 67.5 \sin\left(\frac{\pi}{15} t - \frac{\pi}{2} \right) + 67.5[/latex]
  28. [latex]x(t) = 67.5 \cos\left(\frac{\pi}{15} t - \frac{\pi}{2} \right) = 67.5 \sin\left(\frac{\pi}{15} t \right)[/latex]
  29. [latex]h(t) = 28\sin\left(\frac{2\pi}{3} t - \frac{\pi}{2}\right) + 30[/latex]
  30. .
    1. [latex]\theta(t) = \theta_{0} \sin\left(\sqrt{\frac{g}{l}}\, t + \frac{\pi}{2}\right)[/latex]
    2. [latex]\theta(t) = \frac{\pi}{12} \sin\left(4\pi t + \frac{\pi}{2}\right)[/latex]

Section 7.4 Answers

  1. [latex]\sin(\theta) = \frac{3}{5}, \cos(\theta) = \frac{4}{5}, \tan(\theta) = \frac{3}{4}, \csc(\theta) = \frac{5}{3}, \sec(\theta) = \frac{5}{4}, \cot(\theta) = \frac{4}{3}[/latex]
  2. [latex]\sin(\theta) = \frac{12}{13}, \cos(\theta) = \frac{5}{13}, \tan(\theta) = \frac{12}{5}, \csc(\theta) = \frac{13}{12}, \sec(\theta) = \frac{13}{5}, \cot(\theta) = \frac{5}{12}[/latex]
  3. [latex]\sin(\theta) = \frac{24}{25}, \cos(\theta) = \frac{7}{25}, \tan(\theta) = \frac{24}{7}, \csc(\theta) = \frac{25}{24}, \sec(\theta) = \frac{25}{7}, \cot(\theta) = \frac{7}{24}[/latex]
  4. [latex]\sin(\theta) = \frac{4\sqrt{3}}{7}, \cos(\theta) = \frac{1}{7}, \tan(\theta) = 4\sqrt{3}, \csc(\theta) = \frac{7\sqrt{3}}{12}, \sec(\theta) = 7, \cot(\theta) = \frac{\sqrt{3}}{12}[/latex]
  5. [latex]\sin(\theta) = \frac{\sqrt{91}}{10}, \cos(\theta) = \frac{3}{10}, \tan(\theta) = \frac{\sqrt{91}}{3}, \csc(\theta) = \frac{10\sqrt{91}}{91}, \sec(\theta) = \frac{10}{3}, \cot(\theta) = \frac{3\sqrt{91}}{91}[/latex]
  6. [latex]\sin(\theta) = \frac{\sqrt{530}}{530}, \cos(\theta) = \frac{23\sqrt{530}}{530}, \tan(\theta) = \frac{1}{23}, \csc(\theta) = \sqrt{530}, \sec(\theta) = \frac{\sqrt{530}}{23}, \cot(\theta) = 23[/latex]
  7. [latex]\sin(\theta) = \frac{2\sqrt{5}}{5}, \cos(\theta) = \frac{\sqrt{5}}{5}, \tan(\theta) = 2, \csc(\theta) = \frac{\sqrt{5}}{2}, \sec(\theta) = \sqrt{5}, \cot(\theta) = \frac{1}{2}[/latex]
  8. [latex]\sin(\theta) = \frac{\sqrt{15}}{4}, \cos(\theta) = \frac{1}{4}, \tan(\theta) = \sqrt{15}, \csc(\theta) = \frac{4\sqrt{15}}{15}, \sec(\theta) = 4, \cot(\theta) = \frac{\sqrt{15}}{15}[/latex]
  9. [latex]\sin(\theta) = \frac{\sqrt{6}}{6}, \cos(\theta) = \frac{\sqrt{30}}{6}, \tan(\theta) = \frac{\sqrt{5}}{5}, \csc(\theta) = \sqrt{6}, \sec(\theta) = \frac{\sqrt{30}}{5}, \cot(\theta) = \sqrt{5}[/latex]
  10. [latex]\sin(\theta) = \frac{2\sqrt{2}}{3}, \cos(\theta) = \frac{1}{3}, \tan(\theta) = 2\sqrt{2}, \csc(\theta) = \frac{3\sqrt{2}}{4}, \sec(\theta) = 3, \cot(\theta) = \frac{\sqrt{2}}{4}[/latex]
  11. [latex]\sin(\theta) = \frac{\sqrt{5}}{5}, \cos(\theta) = \frac{2\sqrt{5}}{5}, \tan(\theta) = \frac{1}{2}, \csc(\theta) = \sqrt{5}, \sec(\theta) = \frac{\sqrt{5}}{2}, \cot(\theta) = 2[/latex]
  12. [latex]\sin(\theta) = \frac{1}{5}, \cos(\theta) = \frac{2\sqrt{6}}{5}, \tan(\theta) = \frac{\sqrt{6}}{12}, \csc(\theta) = 5, \sec(\theta) = \frac{5\sqrt{6}}{12}, \cot(\theta) = 2\sqrt{6}[/latex]
  13. [latex]\sin(\theta) = \frac{\sqrt{110}}{11}, \cos(\theta) = \frac{\sqrt{11}}{11}, \tan(\theta) = \sqrt{10}, \csc(\theta) = \frac{\sqrt{110}}{10}, \sec(\theta) = \sqrt{11}, \cot(\theta) = \frac{\sqrt{10}}{10}[/latex]
  14. [latex]\sin(\theta) = \frac{\sqrt{95}}{10}, \cos(\theta) = \frac{\sqrt{5}}{10}, \tan(\theta) = \sqrt{19}, \csc(\theta) = \frac{2\sqrt{95}}{19}, \sec(\theta) = 2\sqrt{5}, \cot(\theta) = \frac{\sqrt{19}}{19}[/latex]
  15. [latex]\sin(\theta) = \frac{\sqrt{21}}{5}, \cos(\theta) = \frac{2}{5}, \tan(\theta) = \frac{\sqrt{21}}{2}, \csc(\theta) = \frac{5\sqrt{21}}{21}, \sec(\theta) = \frac{5}{2}, \cot(\theta) = \frac{2\sqrt{21}}{21}[/latex]
  16. The tree is about 47 feet tall.
  17. The lights are about 75 feet apart.
  18. .
    1. Answer May Vary
    2. The fire is about 4581 feet from the base of the tower.
    3. The Sasquatch ran [latex]200\cot(6^{\circ}) - 200\cot(6.5^{\circ}) \approx 147[/latex] feet in those 10 seconds. This translates to [latex]\approx 10[/latex] miles per hour. At the scene of the second sighting, the Sasquatch was [latex]\approx 1755[/latex] feet from the tower, which means, if it keeps up this pace, it will reach the tower in about 2 minutes.
  19. The tree is about 41 feet tall.
  20. The boat has traveled about 244 feet.
  21. The tower is about 682 feet tall. The guy wire hits the ground about 731 feet away from the base of the tower.
  22. [latex]\tan \left( \dfrac{\pi}{4} \right) = 1[/latex]
  23. [latex]\sec \left( \dfrac{\pi}{6} \right) = \dfrac{2\sqrt{3}}{3}[/latex]
  24. [latex]\csc \left( \dfrac{5\pi}{6} \right) = 2[/latex]
  25. [latex]\cot \left( \dfrac{4\pi}{3} \right) = \dfrac{\sqrt{3}}{3}[/latex]
  26. [latex]\tan \left( -\dfrac{11\pi}{6} \right) = \dfrac{\sqrt{3}}{3}[/latex]
  27. [latex]\sec \left( -\dfrac{3\pi}{2} \right)[/latex] is undefined
  28. [latex]\csc \left( -\dfrac{\pi}{3} \right) = -\dfrac{2\sqrt{3}}{3}[/latex]
  29. [latex]\cot \left( \dfrac{13\pi}{2} \right) = 0[/latex]
  30. [latex]\tan \left( 117\pi \right) = 0[/latex]
  31. [latex]\sec \left( -\dfrac{5\pi}{3} \right) = 2[/latex]
  32. [latex]\csc \left( 3\pi \right)[/latex] is undefined
  33. [latex]\cot \left( -5\pi \right)[/latex] is undefined
  34. [latex]\tan \left( \dfrac{31\pi}{2} \right)[/latex] is undefined
  35. [latex]\sec \left( \dfrac{\pi}{4} \right) = \sqrt{2}[/latex]
  36. [latex]\csc \left( -\dfrac{7\pi}{4} \right) = \sqrt{2}[/latex]
  37. [latex]\cot \left( \dfrac{7\pi}{6} \right) = \sqrt{3}[/latex]
  38. [latex]\tan \left( \dfrac{2\pi}{3} \right) = -\sqrt{3}[/latex]
  39. [latex]\sec \left( -7\pi \right) = -1[/latex]
  40. [latex]\csc \left( \dfrac{\pi}{2} \right) = 1[/latex]
  41. [latex]\cot \left( \dfrac{3\pi}{4} \right) = -1[/latex]
  42. Quadrant II.
  43. Quadrant III.
  44. Quadrant I.
  45. Quadrant IV.
  46. [latex]\sin(\theta) = \frac{3}{5}, \cos(\theta) = -\frac{4}{5}, \tan(\theta) = -\frac{3}{4}, \csc(\theta) = \frac{5}{3}, \sec(\theta) = -\frac{5}{4}, \cot(\theta) = -\frac{4}{3}[/latex]
  47. [latex]\sin(\theta) = -\frac{12}{13}, \cos(\theta) = -\frac{5}{13}, \tan(\theta) = \frac{12}{5}, \csc(\theta) = -\frac{13}{12}, \sec(\theta) = -\frac{13}{5}, \cot(\theta) = \frac{5}{12}[/latex]
  48. [latex]\sin(\theta) = \frac{24}{25}, \cos(\theta) = \frac{7}{25}, \tan(\theta) = \frac{24}{7}, \csc(\theta) = \frac{25}{24}, \sec(\theta) = \frac{25}{7}, \cot(\theta) = \frac{7}{24}[/latex]
  49. [latex]\sin(\theta) = \frac{-4\sqrt{3}}{7}, \cos(\theta) = \frac{1}{7}, \tan(\theta) = -4\sqrt{3}, \csc(\theta) = -\frac{7\sqrt{3}}{12}, \sec(\theta) = 7, \cot(\theta) = -\frac{\sqrt{3}}{12}[/latex]
  50. [latex]\sin(\theta) = -\frac{\sqrt{91}}{10}, \cos(\theta) = -\frac{3}{10}, \tan(\theta) = \frac{\sqrt{91}}{3}, \csc(\theta) = -\frac{10\sqrt{91}}{91}, \sec(\theta) = -\frac{10}{3}, \cot(\theta) = \frac{3\sqrt{91}}{91}[/latex]
  51. [latex]\sin(\theta) = \frac{\sqrt{530}}{530}, \cos(\theta) = -\frac{23\sqrt{530}}{530}, \tan(\theta) = -\frac{1}{23}, \csc(\theta) = \sqrt{530}, \sec(\theta) = -\frac{\sqrt{530}}{23}, \cot(\theta) = -23[/latex]
  52. [latex]\sin(\theta) = -\frac{2\sqrt{5}}{5}, \cos(\theta) = \frac{\sqrt{5}}{5}, \tan(\theta) = -2, \csc(\theta) = -\frac{\sqrt{5}}{2}, \sec(\theta) = \sqrt{5}, \cot(\theta) = -\frac{1}{2}[/latex]
  53. [latex]\sin(\theta) = \frac{\sqrt{15}}{4}, \cos(\theta) = -\frac{1}{4}, \tan(\theta) = -\sqrt{15}, \csc(\theta) = \frac{4\sqrt{15}}{15}, \sec(\theta) = -4, \cot(\theta) = -\frac{\sqrt{15}}{15}[/latex]
  54. [latex]\sin(\theta) = -\frac{\sqrt{6}}{6}, \cos(\theta) = -\frac{\sqrt{30}}{6}, \tan(\theta) = \frac{\sqrt{5}}{5}, \csc(\theta) = -\sqrt{6}, \sec(\theta) = -\frac{\sqrt{30}}{5}, \cot(\theta) = \sqrt{5}[/latex]
  55. [latex]\sin(\theta) = \frac{2\sqrt{2}}{3}, \cos(\theta) = \frac{1}{3}, \tan(\theta) = 2\sqrt{2}, \csc(\theta) = \frac{3\sqrt{2}}{4}, \sec(\theta) = 3, \cot(\theta) = \frac{\sqrt{2}}{4}[/latex]
  56. [latex]\sin(\theta) = \frac{\sqrt{5}}{5}, \cos(\theta) = \frac{2\sqrt{5}}{5}, \tan(\theta) = \frac{1}{2}, \csc(\theta) = \sqrt{5}, \sec(\theta) = \frac{\sqrt{5}}{2}, \cot(\theta) = 2[/latex]
  57. [latex]\sin(\theta) = \frac{1}{5}, \cos(\theta) = -\frac{2\sqrt{6}}{5}, \tan(\theta) = -\frac{\sqrt{6}}{12}, \csc(\theta) = 5, \sec(\theta) = -\frac{5\sqrt{6}}{12}, \cot(\theta) = -2\sqrt{6}[/latex]
  58. [latex]\sin(\theta) = -\frac{\sqrt{110}}{11}, \cos(\theta) = -\frac{\sqrt{11}}{11}, \tan(\theta) = \sqrt{10}, \csc(\theta) = -\frac{\sqrt{110}}{10}, \sec(\theta) = -\sqrt{11}, \cot(\theta) = \frac{\sqrt{10}}{10}[/latex]
  59. [latex]\sin(\theta) = -\frac{\sqrt{95}}{10}, \cos(\theta) = \frac{\sqrt{5}}{10}, \tan(\theta) = -\sqrt{19}, \csc(\theta) = -\frac{2\sqrt{95}}{19}, \sec(\theta) = 2\sqrt{5}, \cot(\theta) = -\frac{\sqrt{19}}{19}[/latex]
  60. [latex]\csc(78.95^{\circ}) \approx 1.019[/latex]
  61. [latex]\tan(-2.01) \approx 2.129[/latex]
  62. [latex]\cot(392.994) \approx 3.292[/latex]
  63. [latex]\sec(207^{\circ}) \approx -1.122[/latex]
  64. [latex]\csc(5.902) \approx -2.688[/latex]
  65. [latex]\tan(39.672^{\circ}) \approx 0.829[/latex]
  66. [latex]\cot(3^{\circ}) \approx 19.081[/latex]
  67. [latex]\sec(0.45) \approx 1.111[/latex]
  68. [latex]\tan(\theta) = \sqrt{3}[/latex] when [latex]\theta = \dfrac{\pi}{3} + \pi k[/latex] for any integer [latex]k[/latex]
  69. [latex]\sec(\theta) = 2[/latex] when [latex]\theta = \dfrac{\pi}{3} + 2\pi k[/latex] or [latex]\theta = \dfrac{5\pi}{3} + 2\pi k[/latex] for any integer [latex]k[/latex]
  70. [latex]\csc(\theta) = -1[/latex] when [latex]\theta = \dfrac{3\pi}{2} + 2\pi k[/latex] for any integer [latex]k[/latex].
  71. [latex]\cot(\theta) = \dfrac{\sqrt{3}}{3}[/latex] when [latex]\theta = \dfrac{\pi}{3} + \pi k[/latex] for any integer [latex]k[/latex]
  72. [latex]\tan(\theta) = 0[/latex] when [latex]\theta = \pi k[/latex] for any integer [latex]k[/latex]
  73. [latex]\sec(\theta) = 1[/latex] when [latex]\theta = 2\pi k[/latex] for any integer [latex]k[/latex]
  74. [latex]\csc(\theta) = 2[/latex] when [latex]\theta = \dfrac{\pi}{6} + 2\pi k[/latex] or [latex]\theta = \dfrac{5\pi}{6} + 2\pi k[/latex] for any integer [latex]k[/latex].
  75. [latex]\cot(\theta) = 0[/latex] when [latex]\theta = \dfrac{\pi}{2} + \pi k[/latex] for any integer [latex]k[/latex]
  76. [latex]\tan(\theta) = -1[/latex] when [latex]\theta = \dfrac{3\pi}{4} + \pi k[/latex] for any integer [latex]k[/latex]
  77. [latex]\sec(\theta) = 0[/latex] never happens
  78. [latex]\csc(\theta) = -\dfrac{1}{2}[/latex] never happens
  79. [latex]\sec(\theta) = -1[/latex] when [latex]\theta = \pi + 2\pi k = (2k+1)\pi[/latex] for any integer [latex]k[/latex]
  80. [latex]\tan(\theta) = -\sqrt{3}[/latex] when [latex]\theta = \dfrac{2\pi}{3} + \pi k[/latex] for any integer [latex]k[/latex]
  81. [latex]\csc(\theta) = -2[/latex] when [latex]\theta = \dfrac{7\pi}{6} + 2\pi k[/latex] or [latex]\theta = \dfrac{11\pi}{6} + 2\pi k[/latex] for any integer [latex]k[/latex]
  82. [latex]\cot(\theta) = -1[/latex] when [latex]\theta = \dfrac{3\pi}{4} + \pi k[/latex] for any integer [latex]k[/latex]
  83. [latex]\cot(t) = 1[/latex] when [latex]t = \dfrac{\pi}{4} + \pi k[/latex] for any integer [latex]k[/latex]
  84. [latex]\tan(t) = \dfrac{\sqrt{3}}{3}[/latex] when [latex]t = \dfrac{\pi}{6} + \pi k[/latex] for any integer [latex]k[/latex]
  85. [latex]\sec(t) = -\dfrac{2\sqrt{3}}{3}[/latex] when [latex]t = \dfrac{5\pi}{6} + 2\pi k[/latex] or [latex]t = \dfrac{7\pi}{6} + 2\pi k[/latex] for any integer [latex]k[/latex]
  86. [latex]\csc(t) = 0[/latex] never happens
  87. [latex]\cot(t) = -\sqrt{3}[/latex] when [latex]t = \dfrac{5\pi}{6} + \pi k[/latex] for any integer [latex]k[/latex]
  88. [latex]\tan(t) = -\dfrac{\sqrt{3}}{3}[/latex] when [latex]t = \dfrac{5\pi}{6} + \pi k[/latex] for any integer [latex]k[/latex]
  89. [latex]\sec(t) = \dfrac{2\sqrt{3}}{3}[/latex] when [latex]t = \dfrac{\pi}{6} + 2\pi k[/latex] or [latex]t = \dfrac{11\pi}{6} + 2\pi k[/latex] for any integer [latex]k[/latex]
  90. [latex]\csc(t) = \dfrac{2\sqrt{3}}{3}[/latex] when [latex]t = \dfrac{\pi}{3} + 2\pi k[/latex] or [latex]t = \dfrac{2\pi}{3} + 2\pi k[/latex] for any integer [latex]k[/latex]
  91. One solution is [latex]g(t) = 3t^2[/latex] and [latex]h(t) = 2\tan(3t)[/latex].
  92. One solution is [latex]g(\theta) = \sec(\theta)[/latex] and [latex]h(\theta) = \tan(\theta)[/latex].
  93. One solution is [latex]g(t) = -\csc(t)[/latex] and [latex]h(t) = \cot(t)[/latex].
  94. One solution is [latex]f(t) = \tan(3t)[/latex] and [latex]g(t) = t[/latex].
  95. One solution is [latex]f(\theta) = 4 \theta[/latex] and [latex]g(\theta) = \tan(\theta)[/latex].
  96. As [latex]\sec^{2}(\theta) = (\sec(\theta))^2[/latex], one solution is [latex]f(\theta) = \sec(\theta)[/latex] and [latex]g(\theta) = \theta^2[/latex].
  97. One solution is [latex]f(x) = \sin(x)[/latex] and [latex]g(x) = \ln(x)[/latex].
  98. One solution is [latex]f(\theta) = \sec(\theta)[/latex], [latex]g(\theta) = \tan(\theta)[/latex], and [latex]h(\theta) = \ln| \theta|[/latex].
  99. Answer May Vary
  100. As we zoom in towards 0, the average rate of change of [latex]\tan(k t)[/latex] approaches [latex]k[/latex]. This is the same trend we observed for [latex]\sin(k t)[/latex] in Section 7.2 number 65. A table with five rows and 4 columns. The first row is the header row with T(t), [-0.1, 0.1], [-0.01, 0.01], and [-0.001, 0.001].  The first column has the functions tangent of t, tangent of 2t, tangent of 3t, and tangent of 4t.  The remaining entries are the values of the functions over the intervals.

Section 7.5 Answers

  1. [latex]y = \tan \left(t - \dfrac{\pi}{3} \right)[/latex]
    Period: [latex]\pi[/latex]
    A single cycle of a tangent curve that goes through (pi over 3, 0) and has asymptotes at t = negative pi over 6 and t = pi over 6.
    Answer to Exercise 1
  2. [latex]y = 2\tan \left( \dfrac{1}{4}t \right) - 3[/latex]
    Period: [latex]4\pi[/latex]
    A single cycle of a tangent curve that goes through (0, -3) and has asymptotes at t = negative 2 pi and t = 2 pi.
    Answer to Exercise 2
  3. [latex]y = \dfrac{1}{3}\tan(-2t - \pi) + 1[/latex] is equivalent to [latex]y = -\dfrac{1}{3}\tan(2t + \pi) + 1[/latex] via the Even / Odd identity for tangent.
    Period: [latex]\dfrac{\pi}{2}[/latex]
    A single cycle of a tangent curve that goes through (- pi over 2, 1) and has asymptotes at t = negative 3 pi over 4 and t = negative pi over 4.
    Answer to Exercise 3
  4. [latex]y = \sec \left( t - \frac{\pi}{2} \right)[/latex]
    Start with [latex]y = \cos \left( t - \frac{\pi}{2} \right)[/latex]
    Period: [latex]2\pi[/latex]
    A single cycle of a secant curve that starts at (pi over 2, 1) and ends at (5 pi over 2,1) and has asymptotes at t = pi and t = 2 pi.
    Answer to Exercise 4
  5. [latex]y = -\csc \left( t + \dfrac{\pi}{3} \right)[/latex]
    Start with [latex]y = -\sin \left( t + \dfrac{\pi}{3} \right)[/latex]
    Period: [latex]2\pi[/latex]
    A single cycle of a cosecant curve that goes through (pi over 6, -1) and (7 pi over 6, 1) and has asymptotes at t=- pi over 3, t = 2 pi over 3, and t = 5 pi over 3.
    Answer to Exercise 5
  6. [latex]y = -\dfrac{1}{3} \sec \left( \dfrac{1}{2}t + \dfrac{\pi}{3} \right)[/latex]
    Start with [latex]y = -\dfrac{1}{3}\cos \left( \dfrac{1}{2}t + \dfrac{\pi}{3} \right)[/latex]
    Period: [latex]4\pi[/latex]
    A single cycle of a secant curve that starts at (-2 pi over 3, -1/3) and ends at (10 pi over 3, -1/3) and has asymptotes at t = pi over 3 and t = 7 pi over 3.
    Answer to Exercise 6
  7. [latex]y = \csc (2t - \pi)[/latex]
    Start with [latex]y = \sin(2t - \pi)[/latex]
    Period: [latex]\pi[/latex]
    A single cycle of a cosecant curve that goes through (3 pi over 4, 1) and (5 pi over 4, -1) and has asymptotes at t= pi over 2, t = pi, and t = 3 pi over 2.
    Answer to Exercise 7
  8. [latex]y = \sec(3t - 2\pi) + 4[/latex]
    Start with [latex]y = \cos (3t - 2\pi) + 4[/latex]
    Period: [latex]\dfrac{2\pi}{3}[/latex]
    A single cycle of a secant curve that starts at (2 pi over 3, 5) and ends at (4 pi over 3, 5) and has asymptotes at t = 5 pi over 6 and t = 7 pi over 6.
    Answer to Exercise 8
  9. [latex]y = \csc \left( -t - \dfrac{\pi}{4} \right) - 2[/latex]
    Start with [latex]y = \sin \left( -t - \dfrac{\pi}{4} \right) - 2[/latex]
    Period: [latex]2\pi[/latex]
    A single cycle of a cosecant curve that goes through (pi over 4, -3) and (5 pi over 4, -1) and has asymptotes at t= - pi over 4, t = 3 pi over 4, and t = 7 pi over 4.
    Answer to Exercise 9
  10. [latex]y = \cot \left( t + \dfrac{\pi}{6} \right)[/latex]
    Period: [latex]\pi[/latex]
    A single cycle of a cotangent curve that goes through (pi over 3, 0) and has asymptotes at t = negative pi over 6 and t = 5 pi over 6.
    Answer to Exercise 10
  11. [latex]y = -11\cot \left( \dfrac{1}{5} t \right)[/latex]
    Period: [latex]5\pi[/latex]
    A single cycle of a cotangent curve that goes through (5 pi over 2, 0) and has asymptotes at t = 0 and t = 5 pi.
    Answer to Exercise 11
  12. [latex]y = \dfrac{1}{3} \cot \left( 2t + \dfrac{3\pi}{2} \right) + 1[/latex]
    Period: [latex]\dfrac{\pi}{2}[/latex]
    A single cycle of a cotangent curve that goes through (- pi over 2, 1) and has asymptotes at t = negative 3 pi over 4 and t = negative pi over 4.
    Answer to Exercise 12
  13. [latex]F(t) = 2 \sec(t-\pi)[/latex], [latex]G(t) = 2 \csc \left(t - \frac{\pi}{2} \right)[/latex]
  14. [latex]F(t) = \sec\left( \frac{\pi}{2} t \right) + 1[/latex], [latex]G(t) = \csc\left( \frac{\pi}{2} t + \frac{\pi}{2} \right) + 1[/latex]
  15. [latex]J(t) = -\tan\left(t+ \frac{\pi}{4} \right)[/latex], [latex]K(t) = \cot \left(t - \frac{\pi}{4} \right)[/latex]
  16. [latex]J(t) = \tan\left( \frac{\pi}{4} t \right) + 1[/latex], [latex]K(t) = -\cot\left( \frac{\pi}{4} t + \frac{\pi}{2} \right) + 1[/latex]
  17. .
    1. [latex]\csc\left(t + \frac{\pi}{2} \right) = \sec(t)[/latex] and [latex]\sec\left(t - \frac{\pi}{2} \right) = \csc(t)[/latex].
    2. [latex]f(t) = \sec\left( 2 t - \frac{7\pi}{6} \right) -1 = \csc\left( \left[2 t - \frac{7\pi}{6}\right] + \frac{\pi}{2} \right) -1 = \csc\left( 2 t - \frac{2\pi}{3} \right) -1[/latex], in terms of cosecants.
  18. [latex]f(t) = - \sec\left(2t - \frac{\pi}{6} \right)-1[/latex] and [latex]f(t) = -\csc\left(2t + \frac{\pi}{3} \right) -1[/latex] are two answers

Section 7.6 Answers

  1. [latex]\arcsin \left( -1 \right) = -\dfrac{\pi}{2}[/latex]
  2. [latex]\arcsin \left( -\dfrac{\sqrt{3}}{2} \right) = -\dfrac{\pi}{3}[/latex]
  3. [latex]\arcsin \left( -\dfrac{\sqrt{2}}{2} \right) = -\dfrac{\pi}{4}[/latex]
  4. [latex]\arcsin \left( -\dfrac{1}{2} \right) = -\dfrac{\pi}{6}[/latex]
  5. [latex]\arcsin \left( 0 \right) = 0[/latex]
  6. [latex]\arcsin \left( \dfrac{1}{2} \right) = \dfrac{\pi}{6}[/latex]
  7. [latex]\arcsin \left( \dfrac{\sqrt{2}}{2} \right) = \dfrac{\pi}{4}[/latex]
  8. [latex]\arcsin \left( \dfrac{\sqrt{3}}{2} \right) = \dfrac{\pi}{3}[/latex]
  9. [latex]\arcsin \left( 1 \right) = \dfrac{\pi}{2}[/latex]
  10. [latex]\arccos \left( -1 \right) = \pi[/latex]
  11. [latex]\arccos \left( -\dfrac{\sqrt{3}}{2} \right) = \dfrac{5\pi}{6}[/latex]
  12. [latex]\arccos \left( -\dfrac{\sqrt{2}}{2} \right) = \dfrac{3\pi}{4}[/latex]
  13. [latex]\arccos \left( -\dfrac{1}{2} \right) = \dfrac{2\pi}{3}[/latex]
  14. [latex]\arccos \left( 0 \right) = \dfrac{\pi}{2}[/latex]
  15. [latex]\arccos \left( \dfrac{1}{2} \right) = \dfrac{\pi}{3}[/latex]
  16. [latex]\arccos \left( \dfrac{\sqrt{2}}{2} \right) = \dfrac{\pi}{4}[/latex]
  17. [latex]\arccos \left( \dfrac{\sqrt{3}}{2} \right) = \dfrac{\pi}{6}[/latex]
  18. [latex]\arccos \left( 1 \right) = 0[/latex]
  19. [latex]\arctan \left( -\sqrt{3} \right) = -\dfrac{\pi}{3}[/latex]
  20. [latex]\arctan \left( -1 \right) = -\dfrac{\pi}{4}[/latex]
  21. [latex]\arctan \left( -\dfrac{\sqrt{3}}{3} \right) = -\dfrac{\pi}{6}[/latex]
  22. [latex]\arctan \left( 0 \right) = 0[/latex]
  23. [latex]\arctan \left( \dfrac{\sqrt{3}}{3} \right) = \dfrac{\pi}{6}[/latex]
  24. [latex]\arctan \left( 1 \right) = \dfrac{\pi}{4}[/latex]
  25. [latex]\arctan \left( \sqrt{3} \right) = \dfrac{\pi}{3}[/latex]
  26. [latex]\text{arccot} \left( -\sqrt{3} \right) = \dfrac{5\pi}{6}[/latex]
  27. [latex]\text{arccot} \left( -1 \right) = \dfrac{3\pi}{4}[/latex]
  28. [latex]\text{arccot} \left( -\dfrac{\sqrt{3}}{3} \right) = \dfrac{2\pi}{3}[/latex]
  29. [latex]\text{arccot} \left( 0 \right) = \dfrac{\pi}{2}[/latex]
  30. [latex]\text{arccot} \left( \dfrac{\sqrt{3}}{3} \right) = \dfrac{\pi}{3}[/latex]
  31. [latex]\text{arccot} \left( 1 \right) = \dfrac{\pi}{4}[/latex]
  32. [latex]\text{arccot} \left( \sqrt{3} \right) = \dfrac{\pi}{6}[/latex]
  33. [latex]\text{arcsec} \left( 2 \right) = \dfrac{\pi}{3}[/latex]
  34. [latex]\text{arccsc} \left( 2 \right) = \dfrac{\pi}{6}[/latex]
  35. [latex]\text{arcsec} \left( \sqrt{2} \right) = \dfrac{\pi}{4}[/latex]
  36. [latex]\text{arccsc} \left( \sqrt{2} \right) = \dfrac{\pi}{4}[/latex]
  37. [latex]\text{arcsec} \left( \dfrac{2\sqrt{3}}{3} \right) = \dfrac{\pi}{6}[/latex]
  38. [latex]\text{arccsc} \left( \dfrac{2\sqrt{3}}{3} \right) = \dfrac{\pi}{3}[/latex]
  39. [latex]\text{arcsec} \left( 1 \right) = 0[/latex]
  40. [latex]\text{arccsc} \left( 1 \right) = \dfrac{\pi}{2}[/latex]
  41. [latex]\text{arcsec} \left( -2 \right) = \dfrac{2\pi}{3}[/latex]
  42. [latex]\text{arcsec} \left( -\sqrt{2} \right) = \dfrac{3\pi}{4}[/latex]
  43. [latex]\text{arcsec} \left( -\dfrac{2\sqrt{3}}{3} \right) = \dfrac{5\pi}{6}[/latex]
  44. [latex]\text{arcsec} \left( -1 \right) = \pi[/latex]
  45. [latex]\text{arccsc} \left( -2 \right) = -\dfrac{\pi}{6}[/latex]
  46. [latex]\text{arccsc} \left( -\sqrt{2} \right) = -\dfrac{\pi}{4}[/latex]
  47. [latex]\text{arccsc} \left( -\dfrac{2\sqrt{3}}{3} \right) = -\dfrac{\pi}{3}[/latex]
  48. [latex]\text{arccsc} \left( -1 \right) = -\dfrac{\pi}{2}[/latex]
  49. [latex]\text{arcsec} \left( -2 \right) = \dfrac{4\pi}{3}[/latex]
  50. [latex]\text{arcsec} \left( -\sqrt{2} \right) = \dfrac{5\pi}{4}[/latex]
  51. [latex]\text{arcsec} \left( -\dfrac{2\sqrt{3}}{3} \right) = \dfrac{7\pi}{6}[/latex]
  52. [latex]\text{arcsec} \left( -1 \right) = \pi[/latex]
  53. [latex]\text{arccsc} \left( -2 \right) = \dfrac{7\pi}{6}[/latex]
  54. [latex]\text{arccsc} \left( -\sqrt{2} \right) = \dfrac{5\pi}{4}[/latex]
  55. [latex]\text{arccsc} \left( -\dfrac{2\sqrt{3}}{3} \right) = \dfrac{4\pi}{3}[/latex]
  56. [latex]\text{arccsc} \left( -1 \right) = \dfrac{3\pi}{2}[/latex]
  57. [latex]\sin\left(\arcsin\left(\dfrac{1}{2}\right)\right) = \dfrac{1}{2}[/latex]
  58. [latex]\sin\left(\arcsin\left(-\dfrac{\sqrt{2}}{2}\right)\right) = -\dfrac{\sqrt{2}}{2}[/latex]
  59. [latex]\sin\left(\arcsin\left(\dfrac{3}{5}\right)\right) = \dfrac{3}{5}[/latex]
  60. [latex]\sin\left(\arcsin\left(-0.42\right)\right) = -0.42[/latex]
  61. [latex]\sin\left(\arcsin\left(\dfrac{5}{4}\right)\right)[/latex] is undefined.
  62. [latex]\cos\left(\arccos\left(\dfrac{\sqrt{2}}{2}\right)\right) = \dfrac{\sqrt{2}}{2}[/latex]
  63. [latex]\cos\left(\arccos\left(-\dfrac{1}{2}\right)\right) = -\dfrac{1}{2}[/latex]
  64. [latex]\cos\left(\arccos\left(\dfrac{5}{13}\right)\right) = \dfrac{5}{13}[/latex]
  65. [latex]\cos\left(\arccos\left(-0.998\right)\right) = -0.998[/latex]
  66. [latex]\cos\left(\arccos\left(\pi \right)\right)[/latex] is undefined.
  67. [latex]\tan\left(\arctan\left(-1\right)\right) = -1[/latex]
  68. [latex]\tan\left(\arctan\left(\sqrt{3}\right)\right) = \sqrt{3}[/latex]
  69. [latex]\tan\left(\arctan\left(\dfrac{5}{12}\right)\right) = \dfrac{5}{12}[/latex]
  70. [latex]\tan\left(\arctan\left(0.965\right)\right) = 0.965[/latex]
  71. [latex]\tan\left(\arctan\left( 3\pi \right)\right) = 3\pi[/latex]
  72. [latex]\cot\left(\text{arccot}\left(1\right)\right) = 1[/latex]
  73. [latex]\cot\left(\text{arccot}\left(-\sqrt{3}\right)\right) = -\sqrt{3}[/latex]
  74. [latex]\cot\left(\text{arccot}\left(-\dfrac{7}{24}\right)\right) = -\dfrac{7}{24}[/latex]
  75. [latex]\cot\left(\text{arccot}\left(-0.001\right)\right) = -0.001[/latex]
  76. [latex]\cot\left(\text{arccot}\left( \dfrac{17\pi}{4} \right)\right) = \dfrac{17\pi}{4}[/latex]
  77. [latex]\sec\left(\text{arcsec}\left(2\right)\right) = 2[/latex]
  78. [latex]\sec\left(\text{arcsec}\left(-1\right)\right) = -1[/latex]
  79. [latex]\sec\left(\text{arcsec}\left(\dfrac{1}{2}\right)\right)[/latex] is undefined.
  80. [latex]\sec\left(\text{arcsec}\left(0.75\right)\right)[/latex] is undefined.
  81. [latex]\sec\left(\text{arcsec}\left( 117\pi \right)\right)= 117\pi[/latex]
  82. [latex]\csc\left(\text{arccsc}\left(\sqrt{2}\right)\right) = \sqrt{2}[/latex]
  83. [latex]\csc\left(\text{arccsc}\left(-\dfrac{2\sqrt{3}}{3}\right)\right) = -\dfrac{2\sqrt{3}}{3}[/latex]
  84. [latex]\csc\left(\text{arccsc}\left(\dfrac{\sqrt{2}}{2}\right)\right)[/latex] is undefined.
  85. [latex]\csc\left(\text{arccsc}\left(1.0001\right)\right) = 1.0001[/latex]
  86. [latex]\csc\left(\text{arccsc}\left( \dfrac{\pi}{4} \right)\right)[/latex] is undefined.
  87. [latex]\arcsin\left(\sin\left(\dfrac{\pi}{6}\right) \right) = \dfrac{\pi}{6}[/latex]
  88. [latex]\arcsin\left(\sin\left(-\dfrac{\pi}{3}\right) \right) = -\dfrac{\pi}{3}[/latex]
  89. [latex]\arcsin\left(\sin\left(\dfrac{3\pi}{4}\right) \right) = \dfrac{\pi}{4}[/latex]
  90. [latex]\arcsin\left(\sin\left(\dfrac{11\pi}{6}\right) \right) = -\dfrac{\pi}{6}[/latex]
  91. [latex]\arcsin\left(\sin\left(\dfrac{4\pi}{3}\right) \right) = -\dfrac{\pi}{3}[/latex]
  92. [latex]\arccos\left(\cos\left(\dfrac{\pi}{4}\right) \right) = \dfrac{\pi}{4}[/latex]
  93. [latex]\arccos\left(\cos\left(\dfrac{2\pi}{3}\right) \right) = \dfrac{2\pi}{3}[/latex]
  94. [latex]\arccos\left(\cos\left(\dfrac{3\pi}{2}\right) \right) = \dfrac{\pi}{2}[/latex]
  95. [latex]\arccos\left(\cos\left(-\dfrac{\pi}{6}\right) \right) = \dfrac{\pi}{6}[/latex]
  96. [latex]\arccos\left(\cos\left(\dfrac{5\pi}{4}\right) \right) = \dfrac{3\pi}{4}[/latex]
  97. [latex]\arctan\left(\tan\left(\dfrac{\pi}{3}\right) \right) = \dfrac{\pi}{3}[/latex]
  98. [latex]\arctan\left(\tan\left(-\dfrac{\pi}{4}\right) \right) = -\dfrac{\pi}{4}[/latex]
  99. [latex]\arctan\left(\tan\left(\pi\right) \right) = 0[/latex]
  100. [latex]\arctan\left(\tan\left(\dfrac{\pi}{2}\right) \right)[/latex] is undefined
  101. [latex]\arctan\left(\tan\left(\dfrac{2\pi}{3}\right) \right) = -\dfrac{\pi}{3}[/latex]
  102. [latex]\text{arccot}\left(\cot\left(\dfrac{\pi}{3}\right) \right) = \dfrac{\pi}{3}[/latex]
  103. [latex]\text{arccot}\left(\cot\left(-\dfrac{\pi}{4}\right) \right) = \dfrac{3\pi}{4}[/latex]
  104. [latex]\text{arccot}\left(\cot\left(\pi\right) \right)[/latex] is undefined
  105. [latex]\text{arccot}\left(\cot\left(\dfrac{3\pi}{2}\right) \right) = \dfrac{\pi}{2}[/latex]
  106. [latex]\text{arccot}\left(\cot\left(\dfrac{2\pi}{3}\right) \right) = \dfrac{2\pi}{3}[/latex]
  107. [latex]\text{arcsec}\left(\sec\left(\dfrac{\pi}{4}\right) \right) = \dfrac{\pi}{4}[/latex]
  108. [latex]\text{arcsec}\left(\sec\left(\dfrac{4\pi}{3}\right) \right) = \dfrac{2\pi}{3}[/latex]
  109. [latex]\text{arcsec}\left(\sec\left( \dfrac{5\pi}{6} \right) \right) = \dfrac{5\pi}{6}[/latex]
  110. [latex]\text{arcsec}\left(\sec\left(-\dfrac{\pi}{2} \right) \right)[/latex] is undefined.
  111. [latex]\text{arcsec}\left(\sec\left(\dfrac{5\pi}{3}\right) \right) = \dfrac{\pi}{3}[/latex]
  112. [latex]\text{arccsc}\left(\csc\left(\dfrac{\pi}{6}\right) \right) = \dfrac{\pi}{6}[/latex]
  113. [latex]\text{arccsc}\left(\csc\left(\dfrac{5\pi}{4}\right) \right) = -\dfrac{\pi}{4}[/latex]
  114. [latex]\text{arccsc}\left(\csc\left( \dfrac{2\pi}{3} \right) \right) = \dfrac{\pi}{3}[/latex]
  115. [latex]\text{arccsc}\left(\csc\left(-\dfrac{\pi}{2} \right) \right) = -\dfrac{\pi}{2}[/latex]
  116. [latex]\text{arccsc}\left(\csc\left(\dfrac{11\pi}{6}\right) \right) = -\dfrac{\pi}{6}[/latex]
  117. [latex]\text{arcsec}\left(\sec\left(\dfrac{11\pi}{12}\right) \right) = \dfrac{11\pi}{12}[/latex]
  118. [latex]\text{arccsc}\left(\csc\left(\dfrac{9\pi}{8}\right) \right) = -\dfrac{\pi}{8}[/latex]
  119. [latex]\text{arcsec}\left(\sec\left(\dfrac{\pi}{4}\right) \right) = \dfrac{\pi}{4}[/latex]
  120. [latex]\text{arcsec}\left(\sec\left(\dfrac{4\pi}{3}\right) \right) = \dfrac{4\pi}{3}[/latex]
  121. [latex]\text{arcsec}\left(\sec\left( \dfrac{5\pi}{6} \right) \right) = \dfrac{7\pi}{6}[/latex]
  122. [latex]\text{arcsec}\left(\sec\left(-\dfrac{\pi}{2} \right) \right)[/latex] is undefined.
  123. [latex]\text{arcsec}\left(\sec\left(\dfrac{5\pi}{3}\right) \right) = \dfrac{\pi}{3}[/latex]
  124. [latex]\text{arccsc}\left(\csc\left(\dfrac{\pi}{6}\right) \right) = \dfrac{\pi}{6}[/latex]
  125. [latex]\text{arccsc}\left(\csc\left(\dfrac{5\pi}{4}\right) \right) = \dfrac{5\pi}{4}[/latex]
  126. [latex]\text{arccsc}\left(\csc\left( \dfrac{2\pi}{3} \right) \right) = \dfrac{\pi}{3}[/latex]
  127. [latex]\text{arccsc}\left(\csc\left(-\dfrac{\pi}{2} \right) \right) = \dfrac{3\pi}{2}[/latex]
  128. [latex]\text{arccsc}\left(\csc\left(\dfrac{11\pi}{6}\right) \right) = \dfrac{7\pi}{6}[/latex]
  129. [latex]\text{arcsec}\left(\sec\left(\dfrac{11\pi}{12}\right) \right) = \dfrac{13\pi}{12}[/latex]
  130. [latex]\text{arccsc}\left(\csc\left(\dfrac{9\pi}{8}\right) \right) = \dfrac{9\pi}{8}[/latex]
  131. [latex]\sin\left(\arccos\left(-\dfrac{1}{2}\right)\right) = \dfrac{\sqrt{3}}{2}[/latex]
  132. [latex]\sin\left(\arccos\left(\dfrac{3}{5}\right)\right) = \dfrac{4}{5}[/latex]
  133. [latex]\sin\left(\arctan\left(-2\right)\right) = -\dfrac{2\sqrt{5}}{5}[/latex]
  134. [latex]\sin\left(\text{arccot}\left(\sqrt{5}\right)\right) = \dfrac{\sqrt{6}}{6}[/latex]
  135. [latex]\sin\left(\text{arccsc}\left(-3\right)\right) = -\dfrac{1}{3}[/latex]
  136. [latex]\cos\left(\arcsin\left(-\dfrac{5}{13}\right)\right) = \dfrac{12}{13}[/latex]
  137. [latex]\cos\left(\arctan\left(\sqrt{7} \right)\right) = \dfrac{\sqrt{2}}{4}[/latex]
  138. [latex]\cos\left(\text{arccot}\left( 3 \right)\right) = \dfrac{3\sqrt{10}}{10}[/latex]
  139. [latex]\cos\left(\text{arcsec}\left( 5 \right)\right) = \dfrac{1}{5}[/latex]
  140. [latex]\tan\left(\arcsin\left(-\dfrac{2\sqrt{5}}{5}\right)\right)=-2[/latex]
  141. [latex]\tan\left(\arccos\left(-\dfrac{1}{2}\right)\right) = -\sqrt{3}[/latex]
  142. [latex]\tan\left(\text{arcsec}\left(\dfrac{5}{3}\right)\right) = \dfrac{4}{3}[/latex]
  143. [latex]\tan\left(\text{arccot}\left( 12 \right)\right) = \dfrac{1}{12}[/latex]
  144. [latex]\cot\left(\arcsin\left(\dfrac{12}{13}\right)\right) = \dfrac{5}{12}[/latex]
  145. [latex]\cot\left(\arccos\left(\dfrac{\sqrt{3}}{2}\right)\right) = \sqrt{3}[/latex]
  146. [latex]\cot\left(\text{arccsc}\left(\sqrt{5}\right)\right) = 2[/latex]
  147. [latex]\cot\left(\arctan \left( 0.25 \right)\right) = 4[/latex]
  148. [latex]\sec\left(\arccos\left(\dfrac{\sqrt{3}}{2}\right)\right) = \dfrac{2\sqrt{3}}{3}[/latex]
  149. [latex]\sec\left(\arcsin\left(-\dfrac{12}{13}\right)\right) = \dfrac{13}{5}[/latex]
  150. [latex]\sec\left(\arctan\left(10\right)\right) = \sqrt{101}[/latex]
  151. [latex]\sec\left(\text{arccot}\left(-\dfrac{\sqrt{10}}{10}\right)\right) = -\sqrt{11}[/latex]
  152. [latex]\csc\left(\text{arccot}\left(9 \right)\right) = \sqrt{82}[/latex]
  153. [latex]\csc\left(\arcsin\left(\dfrac{3}{5}\right)\right) = \dfrac{5}{3}[/latex]
  154. [latex]\csc\left(\arctan\left(-\dfrac{2}{3}\right)\right) = -\dfrac{\sqrt{13}}{2}[/latex]
  155. [latex]\sin\left(\arcsin\left( \dfrac{5}{13} \right) + \dfrac{\pi}{4}\right) = \dfrac{17\sqrt{2}}{26}[/latex]
  156. [latex]\cos\left( \text{arcsec}(3) + \arctan(2) \right) = \dfrac{\sqrt{5} - 4\sqrt{10}}{15}[/latex]
  157. [latex]\tan\left( \arctan(3) + \arccos\left(-\dfrac{3}{5}\right) \right) = \dfrac{1}{3}[/latex]
  158. [latex]\sin\left(2\arcsin\left(-\dfrac{4}{5}\right)\right)= -\dfrac{24}{25}[/latex]
  159. [latex]\sin\left(2\text{arccsc}\left(\dfrac{13}{5}\right)\right) = \dfrac{120}{169}[/latex]
  160. [latex]\sin\left(2\arctan\left(2\right)\right) = \dfrac{4}{5}[/latex]
  161. [latex]\cos\left(2 \arcsin\left(\dfrac{3}{5}\right)\right) = \dfrac{7}{25}[/latex]
  162. [latex]\cos\left(2 \text{arcsec}\left(\dfrac{25}{7}\right)\right) = -\dfrac{527}{625}[/latex]
  163. [latex]\cos\left(2 \text{arccot}\left(-\sqrt{5}\right)\right) = \dfrac{2}{3}[/latex]
  164. [latex]\sin\left( \dfrac{\arctan(2)}{2} \right) = \sqrt{\dfrac{5-\sqrt{5}}{10}}[/latex]
  165. [latex]f(x) = \sin \left( \arccos \left( x \right) \right) = \sqrt{1 - x^{2}}[/latex] for [latex]-1 \leq x \leq 1[/latex]
  166. [latex]f(x) = \cos \left( \arctan \left( x \right) \right) = \dfrac{1}{\sqrt{1 + x^{2}}}[/latex] for all [latex]x[/latex]
  167. [latex]f(x) =\tan \left( \arcsin \left( x \right) \right) = \dfrac{x}{\sqrt{1 - x^{2}}}[/latex] for [latex]-1[/latex] < [latex]x[/latex] < 1
  168. [latex]f(x) =\sec \left( \arctan \left( x \right) \right) = \sqrt{1 + x^{2}}[/latex] for all [latex]x[/latex]
  169. [latex]f(x) =\csc \left( \arccos \left( x \right) \right) = \dfrac{1}{\sqrt{1 - x^{2}}}[/latex] for [latex]-1[/latex] < [latex]x[/latex] < 1
  170. [latex]f(x) =\sin \left( 2\arctan \left( x \right) \right) = \dfrac{2x}{x^{2} + 1}[/latex] for all [latex]x[/latex]
  171. [latex]f(x) =\sin \left( 2\arccos \left( x \right) \right) = 2x\sqrt{1-x^2}[/latex] for [latex]-1 \leq x \leq 1[/latex]
  172. [latex]f(x) =\cos \left( 2\arctan \left( x \right) \right) = \dfrac{1 - x^{2}}{1 + x^{2}}[/latex] for all [latex]x[/latex]
  173.  [latex]f(x) =\sin(\arccos(2x)) = \sqrt{1-4x^2}[/latex] for [latex]-\frac{1}{2} \leq x \leq \frac{1}{2}[/latex]
  174. [latex]f(x) =\sin\left(\arccos\left(\dfrac{x}{5}\right)\right) = \dfrac{\sqrt{25-x^2}}{5}[/latex] for [latex]-5 \leq x \leq 5[/latex]
  175.  [latex]f(x) =\cos\left(\arcsin\left(\dfrac{x}{2}\right)\right) = \dfrac{\sqrt{4-x^2}}{2}[/latex] for [latex]-2 \leq x \leq 2[/latex]
  176. [latex]f(x) =\cos\left(\arctan\left(3x\right)\right) = \dfrac{1}{\sqrt{1+9x^{2}}}[/latex] for all [latex]x[/latex]
  177.  [latex]f(x) =\sin(2\arcsin(7x)) = 14x \sqrt{1-49x^2}[/latex] for [latex]-\dfrac{1}{7} \leq x \leq \dfrac{1}{7}[/latex]
  178. [latex]f(x) =\sin\left(2 \arcsin\left( \dfrac{x\sqrt{3}}{3} \right) \right) = \dfrac{2x\sqrt{3-x^2}}{3}[/latex] for [latex]-\sqrt{3} \leq x \leq \sqrt{3}[/latex]
  179. [latex]f(x) =\cos(2 \arcsin(4x)) = 1 - 32x^2[/latex] for [latex]-\dfrac{1}{4} \leq x \leq \dfrac{1}{4}[/latex]
  180. [latex]f(x) =\sec(\arctan(2x))\tan(\arctan(2x)) = 2x \sqrt{1+4x^2}[/latex] for all [latex]x[/latex]
  181. [latex]f(x) =\sin \left( \arcsin(x) + \arccos(x) \right) = 1[/latex] for [latex]-1 \leq x \leq 1[/latex]
  182. [latex]f(x) =\cos \left( \arcsin(x) + \arctan(x) \right) = \dfrac{\sqrt{1 - x^{2}} - x^{2}}{\sqrt{1 + x^{2}}}[/latex] for [latex]-1 \leq x \leq 1[/latex]
  183. [4] [latex]f(x) =\tan \left( 2\arcsin(x) \right) = \dfrac{2x\sqrt{1 - x^{2}}}{1 - 2x^{2}}[/latex] for [latex]x[/latex] in [latex]\left(-1, -\dfrac{\sqrt{2}}{2}\right) \cup \left(-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2} \right) \cup \left(\dfrac{\sqrt{2}}{2}, 1\right)[/latex]
  184. [latex]f(x) =\sin \left( \dfrac{1}{2}\arctan(x) \right) = \left\{ \begin{array}{rr} \sqrt{\dfrac{\sqrt{x^{2} + 1} - 1}{2\sqrt{x^{2} + 1}}} & \text{for } x \geq 0 \\ & \\ -\sqrt{\dfrac{\sqrt{x^{2} + 1} - 1}{2\sqrt{x^{2} + 1}}} & \text{for } x \text{ } 0 \end{array}\right.[/latex]
  185. [latex]\theta + \sin(2\theta) = \arcsin \left( \dfrac{x}{2} \right) + \dfrac{x\sqrt{4 - x^{2}}}{2}[/latex]
  186. [latex]\dfrac{1}{2}\theta - \dfrac{1}{2}\sin(2\theta) = \dfrac{1}{2} \arctan \left( \dfrac{x}{7} \right) - \dfrac{7x}{x^{2} + 49}[/latex]
  187. [latex]4\tan(\theta) - 4\theta = \sqrt{x^{2} - 16} - 4\mbox{arcsec} \left( \dfrac{x}{4} \right)[/latex]
  188. [latex]\left[-\dfrac{1}{5}, \dfrac{1}{5}\right][/latex]
  189. [latex]\left[-\dfrac{1}{3}, 1 \right][/latex]
  190. [latex]\left[-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right][/latex]
  191. [latex](-\infty, -\sqrt{5}] \cup [-\sqrt{3}, \sqrt{3}] \cup [\sqrt{5}, \infty)[/latex]
  192. [latex](-\infty, \infty)[/latex]
  193. [latex](-\infty, -3) \cup (-3,3) \cup (3, \infty)[/latex]
  194. [latex]\left(\dfrac{1}{2}, \infty \right)[/latex]
  195. [latex]\left[\dfrac{1}{2}, \infty \right)[/latex]
  196. [latex]\left(-\infty, -\dfrac{1}{12}\right] \cup \left[\dfrac{1}{12}, \infty\right)[/latex]
  197. [latex](-\infty, -6] \cup [-4, \infty)[/latex]
  198. [latex](-\infty, -2] \cup [2, \infty)[/latex]
  199. [latex][0, \infty)[/latex]

 

  1. Two cycles of the graph are shown to illustrate the discrepancy discussed. ↵
  2. Again, we graph two cycles to illustrate the discrepancy discussed. ↵
  3. This will be the last time we graph two cycles to illustrate the discrepancy discussed. ↵
  4. The equivalence for [latex]x = \pm 1[/latex] can be verified independently of the derivation of the formula, but Calculus is required to fully understand what is happening at those [latex]x[/latex] values. You'll see what we mean when you work through the details of the identity for [latex]\tan(2t).[/latex] For now, we exclude [latex]x = \pm 1[/latex] from our answer. ↵

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