For the following exercises (1-5), convert each angle in degrees to radians. Write the answer as a multiple of [latex]\pi[/latex].
1. 240°
2. 15°
3. -60°
4. -225°
5. 330°
For the following exercises (6-10), convert each angle in radians to degrees.
6. [latex]\large \frac{\pi}{2}[/latex] rad
7. [latex]\large \frac{7\pi}{6}[/latex] rad
8. [latex]\large\frac{11\pi}{2}[/latex] rad
9. [latex]-3\pi[/latex] rad
10. [latex]\large\frac{5\pi}{12}[/latex] rad
Evaluate the following functional values (11-16).
11. [latex]\cos \Big(\large\frac{4\pi}{3}\Big)[/latex]
12. [latex]\tan \Big(\large\frac{19\pi}{4}\Big)[/latex]
13. [latex]\sin\left(-\large\frac{3\pi}{4}\right)[/latex]
14. [latex]\sec\Big(\large\frac{\pi}{6}\Big)[/latex]
15. [latex]\sin\Big(\large\frac{\pi}{12}\Big)[/latex]
16. [latex]\cos \Big(\large \frac{5\pi}{12}\Big)[/latex]
For the following exercises (17-22), consider triangle [latex]ABC[/latex], a right triangle with a right angle at [latex]C[/latex]. (a) Find the missing side of the triangle, and (b) find the six trigonometric function values for the angle at [latex]A[/latex]. Where necessary, round to one decimal place.
17. [latex]a=4, \, c=7[/latex]
18. [latex]a=21, \, c=29[/latex]
19. [latex]a=85.3, \, b=125.5[/latex]
20. [latex]b=40, \, c=41[/latex]
21. [latex]a=84, \, b=13[/latex]
22. [latex]b=28, \, c=35[/latex]
For the following exercises (23-26), [latex]P[/latex] 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 [latex]\theta [/latex] with a terminal side that passes through point [latex]P[/latex]. Rationalize all denominators.
23. [latex]P\left(\frac{7}{25},y\right), \, y>0[/latex]
24. [latex]P\left(\frac{-15}{17},y\right), \, y<0[/latex]
25. [latex]P\left(x,\frac{\sqrt{7}}{3}\right), \, x<0[/latex]
26. [latex]P\left(x,\frac{−\sqrt{15}}{4}\right), \, x>0[/latex]
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. [latex]\tan^2 x+\sin x\csc x[/latex]
28. [latex]\sec x\sin x\cot x[/latex]
29. [latex]\dfrac{\tan^2 x}{\sec^2 x}[/latex]
30. [latex]\sec x-\cos x[/latex]
31. [latex](1+\tan \theta)^2-2\tan \theta [/latex]
32. [latex]\sin x(\csc x-\sin x)[/latex]
33. [latex]\dfrac{\cos t}{\sin t} + \dfrac{\sin t}{1+\cos t}[/latex]
34. [latex]\dfrac{1+\tan^2 \alpha}{1+\cot^2 \alpha}[/latex]
For the following exercises (35-42), verify that each equation is an identity.
35. [latex]\dfrac{\tan \theta \cot \theta}{\csc \theta} =\sin \theta [/latex]
36. [latex]\dfrac{\sec^2 \theta}{\tan \theta} =\sec \theta \csc \theta [/latex]
37. [latex]\dfrac{\sin t}{\csc t} + \dfrac{\cos t}{\sec t} =1[/latex]
38. [latex]\dfrac{\sin x}{\cos x+1} + \dfrac{\cos x-1}{\sin x} =0[/latex]
39. [latex]\cot \gamma + \tan \gamma = \sec \gamma \csc \gamma [/latex]
40. [latex]\sin^2 \beta + \tan^2 \beta + \cos^2 \beta = \sec^2 \beta [/latex]
41. [latex]\dfrac{1}{1-\sin \alpha} + \dfrac{1}{1+\sin \alpha } =2\sec^2 \alpha [/latex]
42. [latex]\dfrac{\tan \theta -\cot \theta}{\sin \theta \cos \theta} =\sec^2 \theta -\csc^2 \theta [/latex]
For the following exercises (43-50), solve the trigonometric equations on the interval [latex]0\le \theta <2\pi[/latex].
43. [latex]2\sin \theta -1=0[/latex]
44. [latex]1+\cos \theta =\dfrac{1}{2}[/latex]
45. [latex]2\tan^2 \theta =2[/latex]
46. [latex]4\sin^2 \theta -2=0[/latex]
47. [latex]\sqrt{3}\cot \theta +1=0[/latex]
48. [latex]3\sec \theta -2\sqrt{3}=0[/latex]
49. [latex]2\cos \theta \sin \theta =\sin \theta [/latex]
50. [latex]\csc^2 \theta +2\csc \theta +1=0[/latex]
For the following exercises (51-54), each graph is of the form [latex]y=A\sin Bx[/latex] or [latex]y=A\cos Bx[/latex], where [latex]B>0[/latex]. Write the equation of the graph.
For the following exercises (55-60), find (a) the amplitude, (b) the period, and (c) the phase shift with direction for each function.
55. [latex]y=\sin\left(x-\dfrac{\pi}{4}\right)[/latex]
56. [latex]y=3\cos(2x+3)[/latex]
57. [latex]y=\frac{-1}{2}\sin\left(\frac{1}{4}x\right)[/latex]
58. [latex]y=2\cos\left(x-\dfrac{\pi}{3}\right)[/latex]
59. [latex]y=-3\sin(\pi x+2)[/latex]
60. [latex]y=4\cos\left(2x-\dfrac{\pi}{2}\right)[/latex]
61. [T] The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of [latex]120^{\circ}[/latex], how many inches does it move? Approximate to the nearest whole inch.
62. [T] Find the length of the arc intercepted by central angle [latex]\theta [/latex] in a circle of radius [latex]r[/latex]. Round to the nearest hundredth.
a. [latex]r=12.8[/latex] cm, [latex]\theta =\frac{5\pi}{6}[/latex] rad
b. [latex]r=4.378[/latex] cm, [latex]\theta =\frac{7\pi}{6}[/latex] rad
c. [latex]r=0.964[/latex] cm, [latex]\theta =50^{\circ}[/latex]
d. [latex]r=8.55[/latex] cm, [latex]\theta =325^{\circ}[/latex]
63. [T] As a point [latex]P[/latex] moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, [latex]\omega[/latex], and is given by [latex]\omega =\dfrac{\theta}{t}[/latex], where [latex]\theta [/latex] is in radians and [latex]t[/latex] is time. Find the angular speed for the given data. Round to the nearest thousandth.
a. [latex]\theta =\frac{7\pi}{4}[/latex] rad, [latex]t=10[/latex] sec
b. [latex]\theta =\frac{3\pi }{5}[/latex] rad, [latex]t=8[/latex] sec
c. [latex]\theta =\frac{2\pi }{9}[/latex] rad, [latex]t=1[/latex] min
d. [latex]\theta =23.76[/latex] rad, [latex]t=14[/latex] min
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 [latex]x[/latex] is
[latex]\frac{1}{2}x^2 \sin \theta[/latex],
where [latex]\theta [/latex] is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle [latex]\theta =\frac{5\pi}{12}[/latex] rad.
66. [T] A particle travels in a circular path at a constant angular speed [latex]\omega[/latex]. The angular speed is modeled by the function [latex]\omega =9|\cos(\pi t-\frac{\pi}{12})|[/latex]. Determine the angular speed at [latex]t=9[/latex] sec.
67. [T] An alternating current for outlets in a home has voltage given by the function
[latex]V(t)=150\cos 368t[/latex],
where [latex]V[/latex] is the voltage in volts at time [latex]t[/latex] in seconds.
- Find the period of the function and interpret its meaning.
- Determine the number of periods that occur when 1 sec has passed.
68. [T] The number of hours of daylight in a northeast city is modeled by the function
where [latex]t[/latex] 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 [latex]T=50+10\sin\left[\frac{\pi}{12}(t-8)\right][/latex] is a mathematical model of the temperature (in degrees Fahrenheit) at [latex]t[/latex] 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 [latex]T=60^{\circ}[/latex]?
- Sketch the graph of [latex]T[/latex] over [latex]0\le t\le 24[/latex].
70. [T] The function [latex]H(t)=8\sin\left(\frac{\pi}{6}t\right)[/latex] models the height [latex]H[/latex] (in feet) of the tide [latex]t[/latex] hours after midnight. Assume that [latex]t=0[/latex] 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.?