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P R AC T I C E T E S T, p a g e s 6 8 1 – 6 8 4
1. Multiple Choice What are the roots of the equation
2 sin x cos x ⫽ 2 sin2x over the domain 0 ≤ x < 2␲?
␲
5␲
A. x ⫽ 4 , x = 4
7␲
3␲
B. x ⫽ 4 , x = 4
␲
5␲
C. x ⫽ 4 , x = 4 , x ⫽ 0, x ⫽ ␲
3␲
7␲
D. x ⫽ 4 , x = 4 , x ⫽ 0, x ⫽ ␲
2. Multiple Choice What is the simplest form of
cos 5x sin 2x ⫺ sin 5x cos 2x?
A. ⫺sin 3x
B. sin 3x
C. ⫺sin 7x
D. sin 7x
3. Use graphing technology to determine the general solution of the
equation cot x ⫽ sin x ⫹ 1. Give the answers to the nearest
hundredth.
Graph y 1
sin x 1 for 2
tan x
◊ x<2.
The period of the function is 2.
The approximate zeros of the function are: 5.708359, 1.570796,
0.5748263, 4.712389
Alternate zeros have a difference of 2.
The general solution is: x 0.57 2k, k ç or x 4.71 2k, k ç 4. Solve the equation sin x ⫹ 1 ⫽ 2 cos2x over the domain
3␲
␲
- 2 ≤ x < 2 . Give the exact roots.
sin x 1
sin x 1
2 sin2x sin x 1
(2 sin x 1)(sin x 1)
Either 2 sin x 1 0
2 cos2x
2(1 sin2x)
0
0
or sin x 1 0
1
sin x 1
sin x 2
x 7
2
x or x 6
6
7
The roots are: x , x , and x 6
2
6
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Chapter 7: Trigonometric Equations and Identities—Practice Test—Solutions
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1
5. Determine the general solution of the equation sin 4x ⫽ √ over
2
the set of real numbers.
1
2
sin 4x √
4
x
16
3
4
3
x
16
2
The period of sin 4x is
or .
4
2
4x 4x or
So, the general solution is: x 6. For the identity
3
k, k ç or x k, k ç 16
2
16
2
cos u + cot u
⫽ cot u
1 + sin u
a) Determine the non-permissible values of u.
Non-permissible values occur when: sin U 0, U k, k ç or
1 sin U 0
sin U 1
U
k, k ç 2
b) Verify the identity graphically. Sketch or print the graph and
explain how it verifies the identity.
Graph y cos U
sin U
cos U
and y .
sin U
1 sin U
cos U The graphs coincide so the identity is verified.
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␲
c) Verify the identity for u ⫽ 3 . Explain why this verification does
not prove the identity.
Substitute U L.S. in each side of the identity.
3
cos U cot U
1 sin U
cos cot
3
3
1 sin
3
R.S. cot U
cot
3
1
3
√
1
1
a b a√ b
2
3
√
3
1 a b
2
√
32
√
2 3
√
2 3
2
1
√
3
The left side is equal to the right side, so the identity is verified.
This verification does not prove the identity because there may be
values of U, apart from the non-permissible values, for which the left
side does not equal the right side.
d) Prove the identity.
cos U cot U
1 sin U
cos U
cos U sin U
1 sin U
sin U cos U cos U
(sin U)(1 sin U)
L.S. (cos U)(sin U 1)
(sin U)(1 sin U)
cos U
sin U
cot U
R.S.
The left side is equal to the right side, so the identity is proved.
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7. Given angle a in standard position with its terminal arm in
2
Quadrant 3 and sin a ⫽ ⫺3 , and angle b in standard position with
3
its terminal arm in Quadrant 4 and cos b ⫽ 7 , determine each exact
value.
a) cos (a ⫺ b)
b) tan 2a
Use: x2 y2 r2
For angle A, substitute:
y 2, r 3
x2 (2)2 32
√
x— 5
√
x 5 since the terminal
arm of angle A lies in Quadrant 3.
√
So, cos A 5
3
For angle B, substitute:
x 3, r 7
32 y2 72
√
y — 40
√
y 40 since the terminal
arm of angle B lies in Quadrant 4.
√
Substitute for A and B in:
cos (A B)
cos A cos B sin A sin B
sin 2A
cos 2A
2 sin A cos A
cos2A sin2A
√
5
2
2a b a b
3
3
tan 2A √
5 3
40
2
b a b a b a
b
3
7
3
7
√
√
3 5 2 40
21
a
√
8. Prove this identity:
2 2 cos 2U
2 sin 2U
1 cos 2U
sin 2U
1 (1 2 sin2U)
2 sin U cos U
2 sin2U
2 sin U cos U
sin U
cos U
a
√
2
2
5
b a b
3
3
2
√
, or 4 5
2 - 2 cos 2u
sec2u - 1
⫽
2 sin 2u
tan u
L.S. 40
7
So, sin B R.S. sec2U 1
tan U
tan2U
tan U
tan U
sin U
cos U
The left side and the right side simplify to the same expression, so the
identity is proved.
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Chapter 7: Trigonometric Equations and Identities—Practice Test—Solutions
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