3.5 Section Exercises

Verbal

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

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Yes, when the reference angle is $\text{ }\frac{\pi }{4}\text{ }$ and the terminal side of the angle is in quadrants I and III. Thus, at $\text{ }x=\frac{\pi }{4},\frac{5\pi }{4},$ the sine and cosine values are equal.

2. What would you estimate the cosine of $\text{ }\pi \text{ }$ 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?

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Substitute the sine of the angle in for $\text{ }y\text{ }$ in the Pythagorean Theorem $\text{ }{x}^{2}+{y}^{2}=1.\text{ }$ Solve for $\text{ }x\text{ }$ and take the negative solution.

4. Describe the secant function.

5. Tangent and cotangent have a period of $\text{ }\pi .\text{ }$ What does this tell us about the output of these functions?

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The outputs of tangent and cotangent will repeat every $\text{ }\pi \text{ }$ units.

Algebraic

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

6. $\mathrm{tan}\text{ }\frac{\pi }{6}$

7. $\mathrm{sec}\text{ }\frac{\pi }{6}$

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$\frac{2\sqrt{3}}{3}$

8. $\mathrm{csc}\text{ }\frac{\pi }{6}$

9. $\mathrm{cot}\text{ }\frac{\pi }{6}$

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$\sqrt{3}$

10. $\mathrm{tan}\text{ }\frac{\pi }{4}$

11. $\mathrm{sec}\text{ }\frac{\pi }{4}$

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$\sqrt{2}$

12. $\mathrm{csc}\text{ }\frac{\pi }{4}$

13. $\mathrm{cot}\text{ }\frac{\pi }{4}$

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14. $\mathrm{tan}\text{ }\frac{\pi }{3}$

15. $\mathrm{sec}\text{ }\frac{\pi }{3}$

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16. $\mathrm{csc}\text{ }\frac{\pi }{3}$

17. $\mathrm{cot}\text{ }\frac{\pi }{3}$

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$\frac{\sqrt{3}}{3}$

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

18. $\mathrm{tan}\text{ }\frac{5\pi }{6}$

19. $\mathrm{sec}\text{ }\frac{7\pi }{6}$

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$-\frac{2\sqrt{3}}{3}$

20. $\mathrm{csc}\text{ }\frac{11\pi }{6}$

21. $\mathrm{cot}\text{ }\frac{13\pi }{6}$

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\sqrt{3}

$\sqrt{3}$

$\mathrm{tan}\text{ }\frac{7\pi }{4}$

$\mathrm{sec}\text{ }\frac{3\pi }{4}$

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$-\sqrt{2}$

22. $\mathrm{csc}\text{ }\frac{5\pi }{4}$

23. $\mathrm{cot}\text{ }\frac{11\pi }{4}$

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24. $\mathrm{tan}\text{ }\frac{8\pi }{3}$

25. $\mathrm{sec}\text{ }\frac{4\pi }{3}$

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26. $\mathrm{csc}\text{ }\frac{2\pi }{3}$

27. $\mathrm{cot}\text{ }\frac{5\pi }{3}$

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$-\frac{\sqrt{3}}{3}$

28. $\mathrm{tan}\text{ }225°$

29. $\mathrm{sec}\text{ }300°$

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30. $\mathrm{csc}\text{ }150°$

31. $\mathrm{cot}\text{ }240°$

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$\frac{\sqrt{3}}{3}$

32. $\mathrm{tan}\text{ }330°$

33. $\mathrm{sec}\text{ }120°$

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34. $\mathrm{csc}\text{ }210°$

35. $\mathrm{cot}\text{ }315°$

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36. If $\text{ }\text{sin}\text{ }t=\frac{3}{4},$ and $\text{ }\text{ }t\text{ }$ is in quadrant II, find $\text{ }\mathrm{cos}\text{ }t,\mathrm{sec}\text{ }t,\mathrm{csc}\text{ }t,\mathrm{tan}\text{ }t,\mathrm{cot}\text{ }t.$

37. If $\text{ }\text{cos}\text{ }t=-\frac{1}{3},$ and $\text{ }\text{ }t\text{ }$ is in quadrant III, find $\text{ }\mathrm{sin}\text{ }t,\mathrm{sec}\text{ }t,\mathrm{csc}\text{ }t,\mathrm{tan}\text{ }t,\mathrm{cot}\text{ }t.$

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If $\mathrm{sin}\text{ }t=-\frac{2\sqrt{2}}{3},\mathrm{sec}\text{ }t=-3,\mathrm{csc}\text{ }t=-\frac{3\sqrt{2}}{4},\mathrm{tan}\text{ }t=2\sqrt{2},\mathrm{cot}\text{ }t=\frac{\sqrt{2}}{4}$

38. If $\text{ }\mathrm{tan}\text{ }t=\frac{12}{5},$ and $\text{ }0\le t<\frac{\pi }{2},$ find $\text{ }\mathrm{sin}\text{ }t,\mathrm{cos}\text{ }t,\mathrm{sec}\text{ }t,\mathrm{csc}\text{ }t,$ and $\text{ }\mathrm{cot}\text{ }t.$

39..If $\text{ }\mathrm{sin}\text{ }t=\frac{\sqrt{3}}{2}\text{ }$ and $\text{ }\mathrm{cos}\text{ }t=\frac{1}{2},$ find $\text{ }\mathrm{sec}\text{ }t,\mathrm{csc}\text{ }t,\mathrm{tan}\text{ }t,$ and $\text{ }\mathrm{cot}\text{ }t.$

$\mathrm{sec}\text{ }t=2,\mathrm{csc}\text{ }t=\frac{2\sqrt{3}}{3},\text{ }\mathrm{tan}\text{ }t=\sqrt{3},\text{ }\mathrm{cot}\text{ }t=\frac{\sqrt{3}}{3}$

40. If $\text{ }\mathrm{sin}\text{ }40°\approx 0.643\text{ }\text{ }\mathrm{cos}\text{ }40°\approx 0.766\text{ }\text{ }\text{sec}\text{ }40°,\text{csc}\text{ }40°,\text{tan}\text{ }40°,\text{and}\text{ }\text{cot}\text{ }40°.$

If $\text{ }\text{sin}\text{ }t=\frac{\sqrt{2}}{2},$ what is the $\text{ }\text{sin}\left(-t\right)?$

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$\text{ }-\frac{\sqrt{2}}{2}\text{ }$

41. If $\text{ }\text{cos}\text{ }t=\frac{1}{2},$ what is the $\text{ }\text{cos}\left(-t\right)?$

If $\text{ }\text{sec}\text{ }t=3.1,$ what is the $\text{ }\text{sec}\left(-t\right)?$

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42. If $\text{ }\text{csc}\text{ }t=0.34,$ what is the $\text{ }\text{csc}\left(-t\right)?$

43. If $\text{ }\text{tan}\text{ }t=-1.4,$ what is the $\text{ }\text{tan}\left(-t\right)?$

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44. If $\text{ }\text{cot}\text{ }t=9.23,$ what is the $\text{ }\text{cot}\left(-t\right)?$

Graphical

For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.

45.

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.

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$\mathrm{sin}\text{ }t=\frac{\sqrt{2}}{2},\mathrm{cos}\text{ }t=\frac{\sqrt{2}}{2},\mathrm{tan}\text{ }t=1,\mathrm{cot}\text{ }t=1,\mathrm{sec}\text{ }t=\sqrt{2},\mathrm{csc}\text{ }t=\sqrt{2}$

46.

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.

47.

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.

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s i n t = - \frac{\sqrt{3}}{2}, c o s t = - \frac{1}{2}, t a n t = \sqrt{3}, c o t t = \frac{\sqrt{3}}{3}, s e c t = - 2, c s c t = - \frac{2 \sqrt{3}}{3}

$\mathrm{sin}\text{ }t=-\frac{\sqrt{3}}{2},\mathrm{cos}\text{ }t=-\frac{1}{2},\mathrm{tan}\text{ }t=\sqrt{3},\mathrm{cot}\text{ }t=\frac{\sqrt{3}}{3},\mathrm{sec}\text{ }t=-2,\mathrm{csc}\text{ }t=-\frac{2\sqrt{3}}{3}$

Technology

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

48. $\mathrm{csc}\text{ }\frac{5\pi }{9}$

49. $\mathrm{cot}\text{ }\frac{4\pi }{7}$

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50. $\mathrm{sec}\text{ }\frac{\pi }{10}$

51. $\mathrm{tan}\text{ }\frac{5\pi }{8}$

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52. $\mathrm{sec}\text{ }\frac{3\pi }{4}$

53. $\mathrm{csc}\text{ }\frac{\pi }{4}$

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54. $\text{tan}\text{ }98°$

55. $\mathrm{cot}\text{ }33°$

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56. $\mathrm{cot}\text{ }140°$

57. $\mathrm{sec}\text{ }310°$

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Extensions

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

58. If $\text{ }\mathrm{tan}\left(t\right)\approx 2.7,$ and $\text{ }\mathrm{sin}\left(t\right)\approx 0.94,$ find $\text{ }\mathrm{cos}\left(t\right).$

59. If $\text{ }\mathrm{tan}\left(t\right)\approx 1.3,$ and $\text{ }\mathrm{cos}\left(t\right)\approx 0.61,$ find $\text{ }\mathrm{sin}\left(t\right).\text{ }$

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$\mathrm{sin}\left(t\right)\approx 0.79$

60. If $\text{ }\mathrm{csc}\left(t\right)\approx 3.2,$ and $\text{ }\mathrm{cos}\left(t\right)\approx 0.95,$ find $\text{ }\mathrm{tan}\left(t\right).$

61. If $\text{ }\mathrm{cot}\left(t\right)\approx 0.58,$ and $\text{ }\mathrm{cos}\left(t\right)\approx 0.5,$ find $\text{ }\mathrm{csc}\left(t\right).$

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$\mathrm{csc}t\approx 1.16$

62. Determine whether the function $\text{ }f\left(x\right)=2\mathrm{sin}\text{ }x\text{ }\mathrm{cos}\text{ }x\text{ }$ is even, odd, or neither.

63. Determine whether the function $f\left(x\right)=3{\mathrm{sin}}^{2}x\text{ }\mathrm{cos}\text{ }x\text{ }+\text{ }\mathrm{sec}\text{ }x$ is even, odd, or neither.

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64. Determine whether the function $f\left(x\right)=\mathrm{sin}\text{ }x\text{ }-2{\mathrm{cos}}^{2}x$ is even, odd, or neither.

65. Determine whether the function $\text{ }f\left(x\right)={\mathrm{csc}}^{2}x+\mathrm{sec}\text{ }x\text{ }$ is even, odd, or neither.

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

66. $\mathrm{csc}\text{ }t\text{ }\mathrm{tan}\text{ }t$

67. $\frac{\mathrm{sec}\text{ }t}{\mathrm{csc}\text{ }t}$

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$\frac{\mathrm{sin}\text{ }t}{\mathrm{cos}\text{ }t}=\mathrm{tan}\text{ }t$

Real-World Applications

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

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

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13.77 hours, period: $\text{ }1000\pi \text{ }$

70. The equation $\text{ }P=20\mathrm{sin}\left(2\pi t\right)+100\text{ }$ models the blood pressure, $\text{ }P,$ where $\text{ }t\text{ }$ represents time in seconds. (a) Find the blood pressure after 15 seconds. (b) What are the maximum and minimum blood pressures?

71. The height of a piston, $\text{ }h,$ in inches, can be modeled by the equation $\text{ }y=2\mathrm{cos}\text{ }x+6,$ where $\text{ }x\text{ }$ represents the crank angle. Find the height of the piston when the crank angle is $\text{ }55°.\text{ }$

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72. The height of a piston, $\text{ }h,$ in inches, can be modeled by the equation $\text{ }y=2\mathrm{cos}\text{ }x+5,$ where $\text{ }x\text{ }$ represents the crank angle. Find the height of the piston when the crank angle is $\text{ }55°.\text{ }$

Trigonometric Functions