Problem Set: Trigonometric Functions

For the following exercises (1-5), convert each angle in degrees to radians. Write the answer as a multiple of $\pi$ .

1. 240°

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$\frac{4\pi}{3}$ rad

2. 15°

3. -60°

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$\frac{−\pi }{3}$ rad

4. -225°

5. 330°

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$\frac{11\pi}{6}$ rad

For the following exercises (6-10), convert each angle in radians to degrees.

6. $\large \frac{\pi}{2}$ rad

7. $\large \frac{7\pi}{6}$ rad

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8. $\large\frac{11\pi}{2}$ rad

9. $-3\pi$ rad

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10. $\large\frac{5\pi}{12}$ rad

Evaluate the following functional values (11-16).

11. $\cos \Big(\large\frac{4\pi}{3}\Big)$

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12. $\tan \Big(\large\frac{19\pi}{4}\Big)$

13. $\sin\left(-\large\frac{3\pi}{4}\right)$

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

14. $\sec\Big(\large\frac{\pi}{6}\Big)$

15. $\sin\Big(\large\frac{\pi}{12}\Big)$

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$\large \frac{\sqrt{3}-1}{2\sqrt{2}} \normalsize = \large \frac{\sqrt{6}-\sqrt{2}}{4}$

16. $\cos \Big(\large \frac{5\pi}{12}\Big)$

For the following exercises (17-22), consider triangle $ABC$ , a right triangle with a right angle at $C$ . (a) Find the missing side of the triangle, and (b) find the six trigonometric function values for the angle at $A$ . Where necessary, round to one decimal place.

An image of a triangle. The three corners of the triangle are labeled “A”, “B”, and “C”. Between the corner A and corner C is the side b. Between corner C and corner B is the side a. Between corner B and corner A is the side c. The angle of corner C is marked with a right triangle symbol. The angle of corner A is marked with an angle symbol.

17. $a=4, \, c=7$

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a. $b=5.7$ b. $\sin A=\frac{4}{7}, \, \cos A=\frac{5.7}{7}, \, \tan A=\frac{4}{5.7}, \, \csc A=\frac{7}{4}, \, \sec A=\frac{7}{5.7}, \, \cot A=\frac{5.7}{4}$

18. $a=21, \, c=29$

19. $a=85.3, \, b=125.5$

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a. $c=151.7$ b. $\sin A=0.5623, \, \cos A=0.8273, \, \tan A=0.6797, \, \csc A=1.778, \, \sec A=1.209, \, \cot A=1.471$

20. $b=40, \, c=41$

21. $a=84, \, b=13$

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a. $c=85$ b. $\sin A=\frac{84}{85}, \, \cos A=\frac{13}{85}, \, \tan A=\frac{84}{13}, \, \csc A=\frac{85}{84}, \, \sec A=\frac{85}{13}, \, \cot A=\frac{13}{84}$

22. $b=28, \, c=35$

For the following exercises (23-26), $P$ is a point on the unit circle. (a) Find the (exact) missing coordinate value of each point and (b) find the values of the six trigonometric functions for the angle $\theta$ with a terminal side that passes through point $P$ . Rationalize all denominators.

23. $P\left(\frac{7}{25},y\right), \, y>0$

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a. $y=\frac{24}{25}$ b. $\sin \theta =\frac{24}{25}, \, \cos \theta =\frac{7}{25}, \, \tan \theta =\frac{24}{7}, \, \csc \theta =\frac{25}{24}, \, \sec \theta =\frac{25}{7}, \, \cot \theta =\frac{7}{24}$

24. $P\left(\frac{-15}{17},y\right), \, y<0$

25. $P\left(x,\frac{\sqrt{7}}{3}\right), \, x<0$

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a. $x=\frac{−\sqrt{2}}{3}$ b. $\sin \theta =\frac{\sqrt{7}}{3}, \, \cos \theta =\frac{−\sqrt{2}}{3}, \, \tan \theta =\frac{−\sqrt{14}}{2}, \, \csc \theta =\frac{3\sqrt{7}}{7}, \, \sec \theta =\frac{-3\sqrt{2}}{2}, \, \cot \theta =\frac{−\sqrt{14}}{7}$

26. $P\left(x,\frac{−\sqrt{15}}{4}\right), \, x>0$

For the following exercises (27-34), simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

27. $\tan^2 x+\sin x\csc x$

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$\sec^2 x$

28. $\sec x\sin x\cot x$

29. $\dfrac{\tan^2 x}{\sec^2 x}$

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$\sin^2 x$

30. $\sec x-\cos x$

31. $(1+\tan \theta)^2-2\tan \theta$

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$\sec^2 \theta$

32. $\sin x(\csc x-\sin x)$

33. $\dfrac{\cos t}{\sin t} + \dfrac{\sin t}{1+\cos t}$

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$\large\frac{1}{\sin t} \normalsize =\csc t$

34. $\dfrac{1+\tan^2 \alpha}{1+\cot^2 \alpha}$

For the following exercises (35-42), verify that each equation is an identity.

35. $\dfrac{\tan \theta \cot \theta}{\csc \theta} =\sin \theta$

36. $\dfrac{\sec^2 \theta}{\tan \theta} =\sec \theta \csc \theta$

37. $\dfrac{\sin t}{\csc t} + \dfrac{\cos t}{\sec t} =1$

38. $\dfrac{\sin x}{\cos x+1} + \dfrac{\cos x-1}{\sin x} =0$

39. $\cot \gamma + \tan \gamma = \sec \gamma \csc \gamma$

40. $\sin^2 \beta + \tan^2 \beta + \cos^2 \beta = \sec^2 \beta$

41. $\dfrac{1}{1-\sin \alpha} + \dfrac{1}{1+\sin \alpha } =2\sec^2 \alpha$

42. $\dfrac{\tan \theta -\cot \theta}{\sin \theta \cos \theta} =\sec^2 \theta -\csc^2 \theta$

For the following exercises (43-50), solve the trigonometric equations on the interval $0\le \theta <2\pi$ .

43. $2\sin \theta -1=0$

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$\theta = \{\frac{\pi}{6}, \, \frac{5\pi}{6}\}$

44. $1+\cos \theta =\dfrac{1}{2}$

45. $2\tan^2 \theta =2$

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$\theta = \{\frac{\pi}{4}, \, \frac{3\pi}{4}, \, \frac{5\pi}{4}, \, \frac{7\pi}{4}\}$

46. $4\sin^2 \theta -2=0$

47. $\sqrt{3}\cot \theta +1=0$

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$\theta = \{\frac{2\pi}{3}, \, \frac{5\pi}{3}\}$

48. $3\sec \theta -2\sqrt{3}=0$

49. $2\cos \theta \sin \theta =\sin \theta$

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$\theta = \{0, \, \pi, \, \frac{\pi}{3}, \, \frac{5\pi}{3}\}$

50. $\csc^2 \theta +2\csc \theta +1=0$

For the following exercises (51-54), each graph is of the form $y=A\sin Bx$ or $y=A\cos Bx$ , where $B>0$ . Write the equation of the graph.

51.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function that starts at the point (-4, 0) and decreases until the point (-2, 4). After this point the function begins increasing until it hits the point (2, 4). After this point the function begins decreasing again. The x intercepts of the function on this graph are at (-4, 0), (0, 0), and (4, 0). The y intercept is at the origin.

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$y=4\sin\Big(\large\frac{\pi}{4} \normalsize x\Big)$

52.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function that starts at the point (-4, -2) and increases until the point (-3, 2). After this point the function decreases until it hits the point (-2, -2). After this point the function increases until it hits the point (-1, 2). After this point the function decreases until it hits the point (0, -2). After this point the function increases until it hits the point (1, 2). After this point the function decreases until it hits the point (2, -2). After this point the function increases until it hits the point (3, 2). After this point the function begins decreasing again. The x intercepts of the function on this graph are at (-3.5, 0), (-2.5, 0), (-1.5, 0), (-0.5, 0), (0.5, 0), (1.5, 0), (2.5, 0), and (3.5, 0). The y intercept is at the (0, -2).

53.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function. There are many periods and only a few will be explained. The function begins decreasing at the point (-1, 1) and decreases until the point (-0.5, -1). After this point the function increases until it hits the point (0, 1). After this point the function decreases until it hits the point (0.5, -1). After this point the function increases until it hits the point (1, 1). After this point the function decreases again. The x intercepts of the function on this graph are at (-0.75, 0), (-0.25, 0), (0.25, 0), and (0.75, 0). The y intercept is at (0, 1).

Show Solution

$y=\cos(2\pi x)$

54.

For the following exercises (55-60), find (a) the amplitude, (b) the period, and (c) the phase shift with direction for each function.

55. $y=\sin\left(x-\dfrac{\pi}{4}\right)$

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a. 1 b. $2\pi$ c. $\frac{\pi}{4}$ units to the right

56. $y=3\cos(2x+3)$

57. $y=\frac{-1}{2}\sin\left(\frac{1}{4}x\right)$

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a. $\frac{1}{2}$ b. $8\pi$ c. No phase shift

58. $y=2\cos\left(x-\dfrac{\pi}{3}\right)$

59. $y=-3\sin(\pi x+2)$

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a. 3 b. 2 c. $\frac{2}{\pi}$ units to the left

60. $y=4\cos\left(2x-\dfrac{\pi}{2}\right)$

61. [T] The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of $120^{\circ}$ , how many inches does it move? Approximate to the nearest whole inch.

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62. [T] Find the length of the arc intercepted by central angle $\theta$ in a circle of radius $r$ . Round to the nearest hundredth.

a. $r=12.8$ cm, $\theta =\frac{5\pi}{6}$ rad
b. $r=4.378$ cm, $\theta =\frac{7\pi}{6}$ rad
c. $r=0.964$ cm, $\theta =50^{\circ}$
d. $r=8.55$ cm, $\theta =325^{\circ}$

63. [T] As a point $P$ moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, $\omega$ , and is given by $\omega =\dfrac{\theta}{t}$ , where $\theta$ is in radians and $t$ is time. Find the angular speed for the given data. Round to the nearest thousandth.

a. $\theta =\frac{7\pi}{4}$ rad, $t=10$ sec
b. $\theta =\frac{3\pi }{5}$ rad, $t=8$ sec
c. $\theta =\frac{2\pi }{9}$ rad, $t=1$ min
d. $\theta =23.76$ rad, $t=14$ min

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64. [T] A total of 250,000 m² of land is needed to build a nuclear power plant. Suppose it is decided that the area on which the power plant is to be built should be circular.

Find the radius of the circular land area.
If the land area is to form a 45° sector of a circle instead of a whole circle, find the length of the curved side.

65. [T] The area of an isosceles triangle with equal sides of length $x$ is

$\frac{1}{2}x^2 \sin \theta$ ,

where $\theta$ is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle $\theta =\frac{5\pi}{12}$ rad.

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$\approx 30.9 \, \text{in}^2$

66. [T] A particle travels in a circular path at a constant angular speed $\omega$ . The angular speed is modeled by the function $\omega =9|\cos(\pi t-\frac{\pi}{12})|$ . Determine the angular speed at $t=9$ sec.

67. [T] An alternating current for outlets in a home has voltage given by the function

$V(t)=150\cos 368t$ ,

where $V$ is the voltage in volts at time $t$ in seconds.

Find the period of the function and interpret its meaning.
Determine the number of periods that occur when 1 sec has passed.

Show Solution

a. $\pi /184$ ; the voltage repeats every $\pi /184$ sec b. Approximately 59 periods

68. [T] The number of hours of daylight in a northeast city is modeled by the function

N (t) = 12 + 3 \sin [\frac{2 π}{365} (t - 79)]

$N(t)=12+3\sin\left[\dfrac{2\pi}{365}(t-79)\right]$ ,

where $t$ is the number of days after January 1.

Find the amplitude and period.
Determine the number of hours of daylight on the longest day of the year.
Determine the number of hours of daylight on the shortest day of the year.
Determine the number of hours of daylight 90 days after January 1.
Sketch the graph of the function for one period starting on January 1.

69. [T] Suppose that $T=50+10\sin\left[\frac{\pi}{12}(t-8)\right]$ is a mathematical model of the temperature (in degrees Fahrenheit) at $t$ hours after midnight on a certain day of the week.

Determine the amplitude and period.
Find the temperature 7 hours after midnight.
At what time does $T=60^{\circ}$ ?
Sketch the graph of $T$ over $0\le t\le 24$ .

Show Solution

a. Amplitude = 10; period = 24 b. $47.4^{\circ} F$ c. 14 hours later, or 2 p.m. d.

70. [T] The function $H(t)=8\sin\left(\frac{\pi}{6}t\right)$ models the height $H$ (in feet) of the tide $t$ hours after midnight. Assume that $t=0$ is midnight.

Find the amplitude and period.
Graph the function over one period.
What is the height of the tide at 4:30 a.m.?

Functions and Graphs: Supplemental Content

Problem Set: Trigonometric Functions

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