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Honors Precalculus
Chapter 7 Review
1. Use a half-angle identity to find the exact value of sin15°.
2. Verify that cot x =
csc x
sec x
is an identity.
3. Solve for all values of x: cot x sin x + cot x = 0
4. Using the sum and difference identities, find the exact value of cos 105°.
ÍÈÍ π ˙˘˙
4
5. If sin θ = and θ terminates on the interval ÍÍÍÍ 0, ˙˙˙˙ , find the exact value of tan 2θ.
5
ÍÎ 2 ˙˚
6. Solve 4cos 2 x − 3 = 0 for all real values of x.
7. If θ is between 0° and 90° and tanθ =
7
find cosθ .
8
8. If α and β are the measures of two first quadrant angles and sin α =
9. Find the exact value of
4
5
and sin β =
tan155 + tan(−95)
.
1 − tan 155tan(−95)
10. Verify that sin2x = 2cot x sin 2 x is an identity.
11. Verify the identitiy: 1 − 2sinx cos x = (sin x − cos x) 2
12. If cos θ =
3
5
and θ terminates in the first quadrant, find the exact value of sin 2θ.
13. Using the sum and difference identities, find the exact value of sin 15°.
14. Verify that
15. Simplify
1 + cos 2θ
2cos θ
= cos θ is an identity.
1
1
+
.
sinx + 1 sinx − 1
16. Use a half-angle identity to find the exact value of tan 105°.
17. If sin θ =
3
5
and θ terminates in the first quadrant, find the exact value of cos 2θ.
18. Verify that sin (π + θ) = −sin θ is an identity.
19. Find a numerical value of one trigonometric function of x if 5tanx cos x = 4.
20. Verify that sin θ = cos(90° − θ) is an identity.
1
5
13
, find tan(α − β).
21. Find a numerical value of one trigonometric function of x if sinx cot x =
22. If cosθ =
1
3
.
4
and cscθ < 0, find sinθ and tanθ.
7
23. Solve 5tanx = 5 3 for 0° ≤ x ≤ 180°.
24. Simplify (tanx − sec x) (tanx + sec x)
25. If α and β are the measures of two first quadrant angles and sin α =
4
5
and sin β =
5
13
find sin (α + β ).
26. Simplify
tan 10x + tan3x
.
1 − tan10x tan3x
27. Find a numerical value of one trigonometric function of x if sin 2 x − 1 = −1.
28. Which of the following are the solutions of cot 2 x + 2 = 2csc x on the interval [0, 2π)?
29. Find a numerical value of one trigonometric function of x if
tan 2 x − sin 2 x
sin 2 x
= 5.
30. Using the sum and difference identities, find the exact value of tan 105°.
31. Verify that cos 2 x = sec 2 x − tan 2 x − sin 2 x is an identity.
32. Verify that cos 2 x(sec 2 x − 1) = sin 2 x is an identity.
33. Use a half-angle identity to find the exact value of tan 157.5°.
34. Solve th equation for all values of x. 2cot 2 x − 4 = 2
35. Find all solutions of each equation on the interval [0, 360). sin 2 x cos 2 x − 2sin 2 x = 0
36. Solve 2 − 3cos x = 5 + 3cos x for 0° ≤ x ≤ 180°.
37. Verify that cos (270° − θ) = −sin θ is an identity.
38. Verify that (sin x − cos x) 2 = 1 − 2sin x cos x is an identity.
1
39. If sin θ = − and θ terminates in the third quadrant, find the exact value of sin2θ.
4
40. Solve sin2θ = 2cos θ on the interval [0, 2π).
41. Simplify
sec x
sec x
+
.
cos x + 1 cos x − 1
42. Verify sin(180° − θ) = sin θ .
43. Solve sinx + 2sinx cos x = 0 for 0° ≤ x ≤ 180°.
2
,
3
44. If sin θ = − and θ terminates in the fourth quadrant, find the exact value of tan 2θ.
5
45. Solve
tan 2 x
2
− 2cos 2 x = 1 for 0 ≤ x ≤ 2π .
46. Solve tanx sec x − 2tanx = 0 for all real values of x.
7
47. If sin θ = − and θ terminates in the third quadrant, find the exact value of sin2θ .
9
48. Solve cot x + 2cot x sin x = 0 for 0° ≤ x ≤ 180°.
49. Verify the identitiy: sin 2 x = cos 2 x sec 2 x − cos 2 x
3