Math 111 Test 3
!√ "
3
1. Find the exact value of tan−1
.
3
(a) 0
π
(b)
2
π
(c)
6
π
3
7π
(e) −
6
4π
(f) −
3
(d) −
2. Find the exact value of cos−1 (cos 2π)
(a) 2π
(b) π
(c) 0
π
2
π
(e) −
3
(f) None of above
(d) −
3. Solve the equation for x. 50 sin−1 (x) = 25π.
(a) 0
(b) 1
(c) −1
1
(d) −
2
1
(e) −
3
1
(f) −
2
1
4. Find the exact value sec(tan−1 (0))
(a) 1
(b) −1
(c) 0
(d)
1
2
√
2+1
2
√
√
2+2 3
(f)
2
(e)
5. Find the exact value of cos
√
#
cot−1
# $$
1
3
10
(a) −
10
√
(b) 10
√
(c) − 10
√
10
(d)
10
√
(e) − 3
√
(f) 3
6. Write the trigonometric expression tan
x > 0.
#
cos−1
# $$
2
as an algebraic expression in x, x #= 0,
x
(a) x
(b)
(c)
(d)
(e)
√
x2 −4
2
x−2
2
√
x2 −4
x
x−2
x
7. Rewrite the expression
tan θ
cot θ
in terms of sin θ and cos θ, then simplify.
(a) 0
(b)
(c)
sin2 θ
cos2 θ
sin θ
cos θ
(d) − sin θ cos θ
(e) 2 sin θ cos θ
2
8. Simplify the following expression:
sin3 θ + cos3 θ
sin θ + cos θ
(1)
ln | sec θ + tan θ| + ln | sec θ − tan θ|
(1)
(a) 1
(b) −1
(c) 1 − sin θ cos θ
(d) 1 + sin θ cos θ
(e) sin θ cos θ
9. Simplify the following expression:
(a) 1
(b) −1
(c) 0
(d) ln |2 sec θ|
(e) None of above
10. Simplify the following expression:
(2 sin θ cos θ)2 + (cos2 θ − sin2 θ)2
(a) 2 sin2 θ cos2 θ
(b) cos2 θ − 2 sin2 θ
(c) cos2 θ − sin2 θ
(d) −1
(e) 1
11. Find exact value of cos 7π
12
√
2− 6
4
√ √
2+ 6
− 4
√ √
2− 6
4
√ √
2+ 6
4
(a) −
(b)
(c)
(d)
√
(e) None of above
12. If tan(α + β) =
(a)
1
7
1
2
%
&
%
&
and tan α − π4 = − 13 , find exact value of tan β + π4
1
7
(c) −1
(b) −
(d) 1
(e) None of above
3
(1)
13. If cos θ = − 45 . Find exact value of cos(2θ)
(a)
7
25
7
(b) − 25
9
(c) − 25
(d)
9
25
(e) None of above
14. If cos θ = −
(a)
(d)
6
3
and
π
2
< θ < π. Find exact value of sin(2θ)
√
2
3
√
(b) −
(c)
√
2
3
√
2 2
3
√
−232
(e) None of above
15. Find the exact value of tan π8 .
√
(a) 2 + 1
(b)
(c)
(d)
√1
2+1
√1
2−1
√
2−1
(e) None of above
θ
16. Express the product sin 3θ
2 cos 2 as a sum containing only sines or only cosines.
(a) − 12 (sin(2θ) + sin(θ))
(b)
(c)
1
2
1
2
(sin(2θ) + sin(θ))
(cos(2θ) − cos(θ))
(d) − 12 (cos(2θ) − cos(θ))
(e) None of above
17. Determine the total number of solutions of tan(2θ) = 1 for θ in [0, 2π).
(a) 2
(b) 4
(c) 6
(d) 8
(e) 1
4
18. Determine the total number of solutions of sin(2θ − π5 ) =
1
2
for θ in [0, 2π).
(a) 2
(b) 4
(c) 6
(d) 8
(e) 10
19. Solve the equation sin(2θ) − sin(θ) = 0 for θ in [0, 2π), which of the following has ALL the
solutions?
(a)
π
3
(b) − π3
(c)
(d)
(e)
5π
3 ,0
5π
5π
3 ,− 3
0, − 5π
3
(f) None of above
20. Find the exact value of sin
√
(a) − 2
√
(b) 2
√
(c) − 3
√
(d) 3
% 3π &
4
− cos
% 5π &
4
(e) None of above
5
KEY
1. c
2. c
3. b
4. a
5. d
6. b
7. b
8. c
9. c
10. e
11. c
12. d
13. a
14. d
15. d
16. b
17. b
18. b
19. f
20. b
6