Download Exam 2 Study Guide

Exam 2 Study Guide
Math 123: College Trigonometry
March 31, 2016
1. Verify the trigonometric identity: cos2 (x) − sin2 (x) = 2 cos2 (x) − 1
2. Verify the trigonometric identity: sin2 (x) − sin4 (x) = cos2 (x) − cos4 (x)
3. Verify the trigonometric identity: cos(x)(tan2 (x) + 1) = sec(x)
4. Verify the trigonometric identity:
1+sin(x)
cos(x)
+
cos(x)
1+sin(x)
= 2 sec(x)
5. Verify the trigonometric identity:
1
cos(x)+1
+
1
cos(x)−1
= −2 csc(x) cot(x)
6. Verify the trigonometric identity: tan5 (x) = tan3 (x) sec2 (x) − tan3 (x)
7. Verify the trigonometric identity: (sin(x) + cos(x))2 = 1 + sin(2x)
8. Solve: sin(x) =
√
3 − sin(x) in the interval [0, 2π).
9. Solve: 4 cos(x) = 1 + 2 cos(x) in the interval [0, 2π).
10. Solve: 4 tan2 (x) − 1 = tan2 (x) in the interval [0, 2π).
11. Solve: 2 cos2 (x) − cos(x) = 1 in the interval [0, 2π).
12. Solve: 2 cos2 (x) + 3 cos(x) = 0 in the interval [0, 2π).
13. Solve: 2 sin2 (x) + 3 sin(x) + 1 = 0 in the interval [0, 2π).
14. Solve: 2 sin2 (x) = 2 + cos(x) in the interval [0, 2π).
15. Solve: cos2 (x) + sin(x) = 1 in the interval [0, 2π).
16. Solve: sin2 (x) + 2 cos(x) = 2 in the interval [0, 2π).
17. Solve: cos(x) + 1 = sin(x) in the interval [0, 2π).
18. Solve: sec(x) + tan(x) = 1 in the interval [0, 2π).
19. Solve: 2 cos(2x) − 1 = 0 in the interval [0, 2π).
20. Solve: 2 sin(2x) +
√
3 = 0 in the interval [0, 2π).
21. Given t =
a) sin(t)
b) cos(t)
7π
,
12
find the exact values of:(Hint: 7π
=
12
22. Given t =
a) sin(t)
b) cos(t)
−π
,
12
find the exact values of:(Hint: −π
=
12
23. Given sin(x) =
a) sin(x + y)
b) cos(x − y)
c) tan(x + y)
5
13
π
3
+ π4 )
π
6
− π4 )
and cos(y) = − 35 with both x and y are in Quadrant II. Find the exact value for:
7
24. Given sin(x) = − 25
and cos(y) = − 45 with both x and y are in Quadrant III. Find the exact value for:
a) sin(x + y)
b) cos(x + y)
c) tan(x − y)
25. Solve: cos(x + π) − cos(x) − 1 = 0 in the interval [0, 2π).
26. Given that sin(x) = − 35 for
a) sin(2x)
b) cos(2x)
c) tan(2x)
3π
2
27. Given that cos(x) = − 45 for
a) sin(2x)
b) cos(2x)
c) tan(2x)
π
2
< x < 2π. Find the exact values for:
< x < π. Find the exact values for:
28. Given x = π8 , use the half angle formulas to determine the exact values for:
a) sin(x)
b) cos(x)
c) tan(x)
29. Solve: sin( x2 ) + cos(x) = 0 in the interval [0, 2π).
30. Solve: sin( x2 ) + cos(x) − 1 = 0 in the interval [0, 2π).
31. Use the sum-to-product formula to find the exact value of:
cos 75◦ + cos 15◦
32. Use the sum-to-product formula to find the exact value of:
cos 75◦ − cos 15◦
33. Use the product-to-sum formulas to rewrite the product
2 sin(7x) cos(3x)
as a sum or difference.
34. Use the product-to-sum formulas to rewrite the product
cos(4x) sin(6x)
as a sum or difference.