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Pre Calculus B
Review for QUIZ 7.1-7.2, 7.4
Name: _________________________
Date: ________________________
(NO CALCULATOR for tomorrow’s QUIZ)
List the domain and range for the following trigonometric functions so that each function
has an inverse.
y = cos x
y = cos −1 x
y = sin x
y = sin −1 x
y = tan x
y = tan −1 x
True or false?
9π
π
and
______________
8
8
7π
π
The following angles have the same sine value:
______________
and −
6
6
10π
3π
and
The following angles have the same tangent value:
______________
7
7
5π
π
The following angles have the same tangent value:
and − ______________
6
6
9π
7π
and
The following angles have the same cosine value:
______________
8
8
5π
5π
The following angles have the same cosine value: −
______________
and
6
6
The following angles have the same sine value:
Find the exact value of each expression WITHOUT using a calculator.
⎛− 3⎞
⎛ 2⎞
⎛1⎞
1. sin −1 ⎜
2. sin −1 ⎜
3. cos −1 ⎜ ⎟
⎟
⎟
⎜ 2 ⎟
⎜ 2 ⎟
⎝2⎠
⎝
⎠
⎝
⎠
⎛ 2π ⎞
4. cos ⎜
⎟
⎝ 3 ⎠
⎛
3⎞
5. cos −1 ⎜ −
⎜ 2 ⎟⎟
⎝
⎠
6. cos−1 ( 2.532 )
⎛ 4π ⎞
7. sin −1 ⎜ −
⎟
⎝ 3 ⎠
⎛ 3⎞
8. tan −1 ⎜
⎜ 3 ⎟⎟
⎝
⎠
9. tan −1
( 3)
Find the exact value of each expression USING a calculator. State the quadrant that the
angle lies in.
⎛ 3⎞
10. tan −1 (5)
11. cos−1 ( −0.44 )
12. sin −1 ⎜
⎜ 5 ⎟⎟
⎝
⎠
Find the exact value of each composite expression WITHOUT using a calculator.
⎛1⎞
13. sin(sin −1 ⎜ ⎟)
⎝4⎠
14. tan(tan −1 ( 6 ))
⎛2⎞
15. cos(cos −1 ⎜ ⎟)
⎝3⎠
⎛ 9π ⎞
16. sin(sin −1 ⎜
⎟
⎝ 8 ⎠
⎛ −π ⎞
17. sin −1 (sin ⎜
⎟)
⎝ 8 ⎠
⎛ 3π ⎞
18. cos −1 (cos ⎜ ⎟)
⎝ 7 ⎠
⎛ 6π ⎞
21. cos −1 (cos ⎜
⎟)
⎝ 5 ⎠
⎛ ⎛ 2π
19. tan −1 ⎜ tan ⎜ −
⎝ ⎝ 3
⎞⎞
⎟⎟
⎠⎠
⎛ 5π ⎞
20. sin −1 (sin ⎜
⎟)
⎝ 8 ⎠
⎛ 5π
22. cos −1 (cos ⎜ −
⎝ 3
⎞
⎟)
⎠
⎛
⎛ 3π
23. cos −1 ⎜ cos ⎜ −
⎝ 7
⎝
⎛ 5π
25. sin −1 (sin ⎜ −
⎝ 9
⎞⎞
⎟⎟
⎠⎠
⎛ ⎛ 3π
24. tan −1 ⎜ tan ⎜ −
⎝ ⎝ 5
⎞⎞
⎟⎟
⎠⎠
⎞
⎟)
⎠
Find the inverse of the following functions. State the domain and range of f ( x) and
f −1 ( x) .
26. f ( x) = 4sin x − 2
27. f ( x) = 2 tan x + 3
28. f ( x) = −3cos(2 x)
Find the exact value of each composite expression WITHOUT using a calculator.
⎛
3⎞
29. sin ⎜ cos −1
⎟
⎜
2 ⎟⎠
⎝
3π ⎞
⎛
30. sin −1 ⎜ cos
⎟
4 ⎠
⎝
31. cos tan −1 (−1)
⎛ 7π ⎞
32. cos −1 ⎜ sin
⎟
6 ⎠
⎝
⎛
⎛ 1 ⎞⎞
33. cot ⎜ sin −1 ⎜ − ⎟ ⎟
⎝ 2 ⎠⎠
⎝
34. sin tan −1 (−1)
⎛
⎛
2 ⎞⎞
35. csc ⎜ cos −1 ⎜ −
⎜ 2 ⎟⎟ ⎟⎟
⎜
⎝
⎠⎠
⎝
36. cos tan −1 (− 3)
⎛
⎛ 2 ⎞⎞
38. tan ⎜ cos −1 ⎜ ⎟ ⎟
⎝ 5 ⎠⎠
⎝
⎛
⎛ 1 ⎞⎞
39. cos ⎜ sin −1 ⎜ − ⎟ ⎟
⎝ 3 ⎠⎠
⎝
⎛
⎛ 1 ⎞⎞
41. sin ⎜ tan −1 ⎜ ⎟ ⎟
⎝ 4 ⎠⎠
⎝
⎛
⎛ 1 ⎞⎞
42. sec ⎜ tan −1 ⎜ − ⎟ ⎟
⎝ 3 ⎠⎠
⎝
(
)
)
(
)
⎛
⎛ 4 ⎞⎞
37. tan ⎜ sin −1 ⎜ ⎟ ⎟
⎝ 5 ⎠⎠
⎝
⎛
⎛ 2 ⎞⎞
40. cot ⎜ cos −1 ⎜ − ⎟ ⎟
⎝ 5 ⎠⎠
⎝
Establish each identity.
43. (1 − sin 2 θ )(1 + tan 2 θ ) = 1
(
44. 4sin 2 θ + 2cos 2 θ = 4 − 2cos 2 θ
Establish each identity.
sin θ
1 + cos θ
+
= 2csc θ
1 + cos θ
sin θ
45.
tan x − cot x
= 1 − 2cos 2 x
tan x + cot x
46.
47.
1 − sin x
cos3 x
=
sec x
1 + sin x
48. csc x − cot x =
49.
cos x + sin x − sin 3 x
= cot x + cos2 x
sin x
50.
sin x
1 + cos x
sin x + cos x sin x − cos x
−
= sec x csc x
cos x
sin x