Exercises: Trigonometry
1–4 Convert each angle to radians. Express your answers as
fractions of π.
1. 180◦
2. 60◦
3. 45◦
4. 90◦
27. f (x) = cot
29. f (x) =
5–6 Convert each angle to radians. Round your answers to the
nearest hundredth.
5. 24◦
6. 78◦
26. f (x) = csc
x
√ x
π 28. f (x) = cos
x
5
sin x
x
30. f (x) =
31. f (x) = sin2 x cos3 x
33. f (x) =
7–8 Convert each angle to degrees. Round your answers to the
nearest tenth.
7. π/6
25. f (x) = sec 1 − x2
1
x + 3 tan x
32. f (x) = sec x tan x
sin x
1 − cos x
34. f (x) =
1
(sin x + cos x)3
35. f (x) = csc2 (π x)
1
36. f (x) = cot
1−x
37. f (x) = tan2 (x) cot(1.4)
38. f (x) =
8. 0.73
9–10
Convert each rate of change to radians per minute. Round
your answers to the nearest thousandth.
9. 42◦/ hour
39. f (x) =
10. 0.4◦/ sec
√
3
sin 4x
sin(x)
sin(0.2)
1
40. f (x) = sec √
x
11–12 Use your calculator to compute the value of the given
expression, correct to four decimal places.
41. Find the equation of the tangent line to the curve y = cos 4x at
the point (π/12, 1/2).
11. sec2 (0.3)
12. csc(1.2) cot(1.2)
42. In the following triangle, the angle θ is increasing at a rate
of 7.0◦/sec.
13. cot(27◦ )
14. sin−1 (0.6)
15–16 Find the length x. Round your answers to the nearest
hundredth.
2
15.
x
10
16.
x
Θ
(a) How quickly is x increasing when θ = 20◦ ?
(b) How quickly is x increasing when x = 2?
43. In the following triangle, the length x is increasing at a rate of
0.5 units/sec.
0.7
35°
3
x
17–18 Find the angle θ in radians. Round your answers to the
nearest hundredth.
x
Θ
5
6
17.
3
18.
Θ
Θ
5
19–40
How quickly is the angle θ increasing when θ = 30◦ ?
44. In the following triangle, the angle θ is increasing at a rate
of 12◦/min.
4
Compute f 0 (x).
19. f (x) = cos x4
20. f (x) = 1 + sin 5x
21. f (x) = sin3 x
22. f (x) = cos2 (4x)
23. f (x) = x2 tan x
24. f (x) =
√
tan x
3
x
Θ
4
How quickly is x increasing when θ = 40◦ ?
Answers
1. π
2. π/3
11. 1.0957
3. π/4
12. 0.4171
19. −4x3 sin x4
x
1
csc2
5
5
5. 0.42
13. 1.9626
6. 1.36
14. 0.6435
2
20. 15 1 + sin 5x (cos 5x)
23. 2x tan x + x2 sec2 x
27. −
4. π/2
28.
32. sec x tan2 x + sec3 x
24.
sec2 x
√
2 tan x
π sin(π/x)
x2
33.
1
1
2
36. −
csc
(1 − x)2
1−x
7. 30◦
15. 1.15
9. 0.012/min
16. 11.87
21. 3 sin2 x cos x
29.
cos x
sin x
− 2
x
x
37. 2 tan(x) sec2 (x) cot(1.4)
41. y =
42. (a) 0.344 units/sec (b) 0.273 units/sec
30. −
34.
17. 0.59
26.
− csc
1 + 3 sec2 x
(x + 3 tan x)2
3(sin x − cos x)
(sin x + cos x)4
38.
10. 0.419/min
18. 0.64
22. −8 sin(4x) cos(4x)
25. −2x sec 1 − x2 tan 1 − x2
cos x
sin2 x
−
1 − cos x
(1 − cos x)2
1
40. − x−3/2 sec x−1/2 tan x−1/2
2
8. 41.8◦
cos(x)
sin(0.2)
39.
√ √ x cot x
√
2 x
31. 2 sin x cos4 x − 3 sin3 x cos2 x
35. −2π csc2 (π x) cot(π x)
4
(sin 4x)−2/3 (cos 4x)
3
√ 1
π −2 3 x−
2
12
43. 0.15/sec, or 8.6◦/sec
44. 1.428 units/sec