Additional Section Exercises

1. On an interval of [latex]\left[0,2\pi \right)[/latex], can the sine and cosine values of a radian measure ever be equal? If so, where?

2. What would you estimate the cosine of [latex]\pi[/latex] degrees to be? Explain your reasoning.

3. For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?

4. Describe the secant function.

5. Tangent and cotangent have a period of [latex]\pi[/latex]. What does this tell us about the output of these functions?

For the following exercises, find the exact value of each expression.

6. [latex]\tan \frac{\pi }{6}[/latex]

7. [latex]\sec \frac{\pi }{6}[/latex]

8. [latex]\csc \frac{\pi }{6}[/latex]

9. [latex]\cot \frac{\pi }{6}[/latex]

10. [latex]\tan \frac{\pi }{4}[/latex]

11. [latex]\sec \frac{\pi }{4}[/latex]

12. [latex]\csc \frac{\pi }{4}[/latex]

13. [latex]\cot \frac{\pi }{4}[/latex]

14. [latex]\tan \frac{\pi }{3}[/latex]

15. [latex]\sec \frac{\pi }{3}[/latex]

16. [latex]\csc \frac{\pi }{3}[/latex]

17. [latex]\cot \frac{\pi }{3}[/latex]

For the following exercises, use reference angles to evaluate the expression.

18. [latex]\tan \frac{5\pi }{6}[/latex]

19. [latex]\sec \frac{7\pi }{6}[/latex]

20. [latex]\csc \frac{11\pi }{6}[/latex]

21. [latex]\cot \frac{13\pi }{6}[/latex]

22. [latex]\tan \frac{7\pi }{4}[/latex]

23. [latex]\sec \frac{3\pi }{4}[/latex]

24. [latex]\csc \frac{5\pi }{4}[/latex]

25. [latex]\cot \frac{11\pi }{4}[/latex]

26. [latex]\tan \frac{8\pi }{3}[/latex]

27. [latex]\sec \frac{4\pi }{3}[/latex]

28. [latex]\csc \frac{2\pi }{3}[/latex]

29. [latex]\cot \frac{5\pi }{3}[/latex]

30. [latex]\tan 225^\circ[/latex]

31. [latex]\sec 300^\circ[/latex]

32. [latex]\csc 150^\circ[/latex]

33. [latex]\cot 240^\circ[/latex]

34. [latex]\tan 330^\circ[/latex]

35. [latex]\sec 120^\circ[/latex]

36. [latex]\csc 210^\circ[/latex]

37. [latex]\cot 315^\circ[/latex]

38. If [latex]\text{sin}t=\frac{3}{4}[/latex], and [latex]t[/latex] is in quadrant II, find [latex]\cos t,\sec t,\csc t,\tan t,\cot t[/latex].

39. If [latex]\text{cos}t=-\frac{1}{3}[/latex], and [latex]t[/latex] is in quadrant III, find [latex]\sin t,\sec t,\csc t,\tan t,\cot t[/latex].

40. If [latex]\tan t=\frac{12}{5}[/latex], and [latex]0\le t<\frac{\pi }{2}[/latex], find [latex]\sin t,\cos t,\sec t,\csc t[/latex], and [latex]\cot t[/latex]. 41. If [latex]\sin t=\frac{\sqrt{3}}{2}[/latex] and [latex]\cos t=\frac{1}{2}[/latex], find [latex]\sec t,\csc t,\tan t[/latex], and [latex]\cot t[/latex]. 42. If [latex]\sin 40^\circ \approx 0.643\cos 40^\circ \approx 0.766\text{sec}40^\circ ,\text{csc}40^\circ ,\text{tan}40^\circ ,\text{and}\text{cot}40^\circ[/latex]. 43. If [latex]\text{sin}t=\frac{\sqrt{2}}{2}[/latex], what is the [latex]\text{sin}\left(-t\right)?[/latex] 44. If [latex]\text{cos}t=\frac{1}{2}[/latex], what is the [latex]\text{cos}\left(-t\right)?[/latex] 45. If [latex]\text{sec}t=3.1[/latex], what is the [latex]\text{sec}\left(-t\right)?[/latex] 46. If [latex]\text{csc}t=0.34[/latex], what is the [latex]\text{csc}\left(-t\right)?[/latex] 47. If [latex]\text{tan}t=-1.4[/latex], what is the [latex]\text{tan}\left(-t\right)?[/latex] 48. If [latex]\text{cot}t=9.23[/latex], what is the [latex]\text{cot}\left(-t\right)?[/latex] For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions. 49. Graph of circle with angle of t inscribed. Point of (square root of 2 over 2, square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

50.
Graph of circle with angle of t inscribed. Point of (square root of 3 over 2, 1/2) is at intersection of terminal side of angle and edge of circle.

51.
Graph of circle with angle of t inscribed. Point of (-1/2, negative square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.

For the following exercises, use a graphing calculator to evaluate.

52. [latex]\csc \frac{5\pi }{9}[/latex]

53. [latex]\cot \frac{4\pi }{7}[/latex]

54. [latex]\sec \frac{\pi }{10}[/latex]

55. [latex]\tan \frac{5\pi }{8}[/latex]

56. [latex]\sec \frac{3\pi }{4}[/latex]

57. [latex]\csc \frac{\pi }{4}[/latex]

58. [latex]\text{tan}98^\circ[/latex]

59. [latex]\cot 33^\circ[/latex]

60. [latex]\cot 140^\circ[/latex]

61. [latex]\sec 310^\circ[/latex]

For the following exercises, use identities to evaluate the expression.

62. If [latex]\tan \left(t\right)\approx 2.7[/latex], and [latex]\sin \left(t\right)\approx 0.94[/latex], find [latex]\cos \left(t\right)[/latex].

63. If [latex]\tan \left(t\right)\approx 1.3[/latex], and [latex]\cos \left(t\right)\approx 0.61[/latex], find [latex]\sin \left(t\right)[/latex].

64. If [latex]\csc \left(t\right)\approx 3.2[/latex], and [latex]\cos \left(t\right)\approx 0.95[/latex], find [latex]\tan \left(t\right)[/latex].

65. If [latex]\cot \left(t\right)\approx 0.58[/latex], and [latex]\cos \left(t\right)\approx 0.5[/latex], find [latex]\csc \left(t\right)[/latex].

66. Determine whether the function [latex]f\left(x\right)=2\sin x\cos x[/latex] is even, odd, or neither.

67. Determine whether the function [latex]f\left(x\right)=3{\sin }^{2}x\cos x+\sec x[/latex] is even, odd, or neither.

68. Determine whether the function [latex]f\left(x\right)=\sin x - 2{\cos }^{2}x[/latex] is even, odd, or neither.

69. Determine whether the function [latex]f\left(x\right)={\csc }^{2}x+\sec x[/latex] is even, odd, or neither.

For the following exercises, use identities to simplify the expression.

70. [latex]\csc t\tan t[/latex]

71. [latex]\frac{\sec t}{\csc t}[/latex]

72. The amount of sunlight in a certain city can be modeled by the function [latex]h=15\cos \left(\frac{1}{600}d\right)[/latex], where [latex]h[/latex] represents the hours of sunlight, and [latex]d[/latex] is the day of the year. Use the equation to find how many hours of sunlight there are on February 10, the 42nd day of the year. State the period of the function.

73. The amount of sunlight in a certain city can be modeled by the function [latex]h=16\cos \left(\frac{1}{500}d\right)[/latex], where [latex]h[/latex] represents the hours of sunlight, and [latex]d[/latex] is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the 267th day of the year. State the period of the function.

74. The equation [latex]P=20\sin \left(2\pi t\right)+100[/latex] models the blood pressure, [latex]P[/latex], where [latex]t[/latex] represents time in seconds. (a) Find the blood pressure after 15 seconds. (b) What are the maximum and minimum blood pressures?

75. The height of a piston, [latex]h[/latex], in inches, can be modeled by the equation [latex]y=2\cos x+6[/latex], where [latex]x[/latex] represents the crank angle. Find the height of the piston when the crank angle is [latex]55^\circ[/latex].

76. The height of a piston, [latex]h[/latex], in inches, can be modeled by the equation [latex]y=2\cos x+5[/latex], where [latex]x[/latex] represents the crank angle. Find the height of the piston when the crank angle is [latex]55^\circ[/latex].