Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Simplify the expression.
cos θ
1)
+ tan θ
1 + sin θ
A) sec θ
1)
C) cos θ + sin θ
B) sin2 θ
D) 1
Complete the identity.
sin θ cos θ
2)
+ = ?
cos θ sin θ
A) -2 tan2 θ
3) tan4 θ - sec4 θ = ?
A) sec 2 θ + tan 2 θ
2)
B) sec θ csc θ
C) sin θ tan θ
D) 1 + cot θ
3)
C) -2 tan2 θ - 1
B) sec 2 θ
D) tan 2 θ - sec2 θ
Find the exact value of the expression.
11π
4) sin 12
A)
2( 3 - 1)
4)
2( 3 - 1)
4
B)
C) - 5) sin 20° cos 100° + cos 20° sin 100°
1
1
A) - B)
2
3
2( 3 - 1)
4
D) - 2( 3 - 1)
5)
C) - Complete the identity.
6) sin (α + β) cos β - cos (α + β) sin β = ?
A) sin α cos2 β - sin α sin2 β
3
2
D)
3
2
6)
B) sin α
D) sin α cos β - cos α sin β
C) 2 sin β cos β (sin α - cos α )
Find the exact value of the expression.
2
1
7) sin sin-1 + cos-1 3
3
A)
2 6
5
B)
7)
2 3
5
C)
2 + 2 10
9
D)
2 3 + 2 10
9
Use the information given about the angle θ, 0 ≤ θ ≤ 2π, to find the exact value of the indicated trigonometric function.
3
8) csc θ = - , tan θ > 0
Find cos (2θ).
8)
2
A) - 1
9
B)
-4 5
9
C)
1
1
9
D)
4 5
9
9) sin θ = A)
2 6
, tan θ < 0
5
-4 6
25
Find sin (2θ).
B) - 9)
23
25
C)
4 6
25
D)
23
25
Complete the identity.
10) sin θ cos3 θ + sin3 θ cos θ = ?
10)
A)
1
sin2 (2θ)
4
B)
1
sin (2θ) cos (2θ)
2
C)
1
sin (3θ) sin (2θ)
6
D)
1
sin (2θ)
2
Use the information given about the angle θ, 0 ≤ θ ≤ 2π, to find the exact value of the indicated trigonometric function.
3
3π
θ
11) cos θ = - , π < θ < Find cos .
11)
5
2
2
A)
5
5
B) - 3 3π
12) sin θ = - , < θ < 2π
5
2
A) - 10
10
30
10
C)
30
10
D) - 5
5
θ
Find sin .
2
B) - 12)
30
10
C) - 5
5
D)
5
5
Use the Half-angle Formulas to find the exact value of the trigonometric function.
5π
13) cos 12
A) - 1
2
2 - 3
B) - 1
2
2 + 3
C)
1
2
2 - 3
Express the product as a sum containing only sines or cosines.
14) sin (5θ) cos (2θ)
1
1
A) [sin (7θ) + sin (3θ)]
B) [sin (7θ) + cos (3θ)]
2
2
C) sin cos (10θ2 )
D)
D)
1
2
2 + 3
14)
1
[cos (7θ) - cos (3θ)]
2
Express the sum or difference as a product of sines and/or cosines.
15) cos (3θ) - cos (5θ)
A) cos (-2θ)
B) -2 sin (4θ) sin θ
C) -2 cos (4θ) sin θ
D) 2 sin (4θ) sin θ
2
13)
15)
Solve the equation on the interval 0 ≤ θ < 2π.
3
16) sin (4θ) = 2
16)
A) 0
C)
B)
π π 2π 7π 7π 13π 5π 19π
, , , , , , , 12 6 3 12 6
12
3
12
17) 2 cos θ + 1 = 0
3π
A)
2
π 5π
, 4 4
D) 0, π
, π
4
17)
π 5π
B) , 3 3
π 3π
C) , 2 2
2π 4π
D)
, 3
3
Solve the equation. Give a general formula for all the solutions.
2
18) cos (2θ) = 2
π
7π
A) θ = + kπ, θ = + kπ
8
8
C) θ = π
7π
B) θ = + 2kπ, θ = + 2kπ
8
8
4π
2π
+ kπ, θ = + kπ
3
3
19) tan θ = -1
π
A) θ = + 2kπ
4
18)
π
3π
D) θ = + kπ, θ = + kπ
4
4
19)
3π
B) θ = + kπ
4
3π
C) θ = + 2kπ
4
π
D) θ = + kπ
4
π 5π
C) , 6 6
π 5π
D) 0, π, , 6 6
Solve the equation on the interval 0 ≤ θ < 2π.
20) 2 sin2 θ = sin θ
π 2π
A) , 3 3
20)
π 3π π 2π
B) , , , 2 2 3 3
21) sin2 θ - cos2 θ = 0
π π
A) , 4 6
C)
21)
π 3π 5π 7π
B) , , , 4 4
4
4
π
4
D)
22)
22) sin (2θ) + sin θ = 0
π 9π
A) , 8 8
C)
π π
, 4 3
2π
4π
B) 0, , π, 3
3
π 3π 5π 7π
, , , 4 4
4
4
D) No solution
3