Exercises: Trigonometry
1–4 Convert each angle to radians. Express your answers as
fractions of π.
25. f (x) = cos 5x2
26. f (x) = sin
1
x
1. 180◦
2. 60◦
27. f (x) = 2 sin(3x)
28. f (x) = sin(πx) cos(0.2)
3. 45◦
4. 90◦
29. f (x) = sin(cos x)
30. f (x) = sin2 (5x)
5–6 Convert each angle to radians. Round your answers to the
nearest hundredth.
5. 24◦
6. 78◦
Convert each angle to degrees. Round your answers to the
7–10
nearest tenth.
7. π/6
32. Find the equation of the tangent line to the curve y = sin2 x at
the point (π/4, 1/2).
33. In the following triangle, the angle θ is increasing at a rate of
0.13 rad/sec.
8. 3π/2
9. 0.73
31. Find the equation of the tangent line to the curve y = sin x at
the point (0, 0).
10. 3.02
3
11–12 Convert each rate of change to radians per minute. Round
your answers to the nearest thousandth.
11. 42◦/ hour
2
x
Θ
(a) How quickly is x increasing when θ = 0.4?
(b) How quickly is x increasing when x = 2?
12. 0.4◦/ sec
13–14 Find the length x. Round your answers to the nearest
hundredth.
13.
34. In the following triangle, the length x is increasing at a rate of
0.5 units/sec.
x
10
14.
Θ
0.7
35°
5
x
15–16 Find the angle θ in radians. Round your answers to the
nearest hundredth.
How quickly is the angle θ increasing when θ = 30◦ ?
35–36
6
15.
3
16.
Θ
Use the formula
d tan x = 1 + tan2 x to compute f 0 (x).
dx
1
tan x
35. f (x) = x2 tan x
36. f (x) =
37. f (x) = tan(3x)
38. f (x) = tan2 x
Θ
5
17–30
x
4
Compute f 0 (x).
17. f (x) = x2 + 3 sin x
19. f (x) =
sin x
x
√
18. f (x) = x3 cos x
20. f (x) =
x2 + cos x
sin x
22. f (x) = cos3 x
1
23. f (x) =
cos2 x
2
21. f (x) =
39. In the following triangle, the angle θ is increasing at a rate
of 0.2 rad/min.
3
x
Θ
4
24. f (x) = x sin x
How quickly is x increasing when θ = 0.5?
Answers
1. π
2. π/3
3. π/4
12. 0.419 rad/min
19. −
4. π/2
13. 1.15
sin x
cos x
+
x2
x
5. 0.42
14. 11.87
29. −(sin x) cos(cos x)
25. −10x sin 5x2
30. 10 sin(5x) cos(5x)
33. (a) 0.36 units/sec (b) 0.29 units/sec
1 + tan2 x
tan2 x
15. 0.59
37. 3 1 + tan2 (3x)
8. 270◦
21.
26. −
cos x
√
2 sin x
34. 0.15 rad/sec, or 8.6◦ /sec
11. 0.012 rad/min
18. 3x2 cos x − x3 sin x
27. 6 cos(3x)
32. y =
38. 2(tan x) 1 + tan2 x
10. 173.0◦
22. −3 cos2 x (sin x)
cos(1/x)
x2
31. y = x
9. 41.8◦
17. 2x + 3 cos x
16. 0.64
2
20. 3 x2 + cos x ( 2x − sin x )
24. sin2 x + 2x (sin x)(cos x)
36. −
7. 30◦
6. 1.36
23.
2 sin x
cos3 x
28. π cos(πx) cos(0.2)
1
π
+ x−
2
4
35. 2x tan x + x2 1 + tan2 x
39. 1.04 units/min