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Checkpoint: Assess Your Understanding, pages 602–604
7.1
1
1. Multiple Choice How many roots does the equation sin 6x = 3
have over the domain 0 ≤ x < 2␲?
A. 2
B. 4
C. 6
D. 12
2. Use graphing technology to solve each equation over the given
domain. Give the roots to the nearest hundredth.
a) 1 ⫹ 2 sin x ⫽ 1 ⫺ 3 cos x; 0 ≤ x ≤ 2␲
Graph the corresponding function: y ⴝ 2 sin x ⴙ 3 cos x
Determine the approximate zeros in the given domain.
The roots are approximately: x ⴝ 2.16 and x ⴝ 5.30
Substitute each root into the given equation to verify.
b) 2 ⫽ cos x ⫹ 2 cos2x; ⫺2␲ ≤ x ≤ 2␲
Graph the corresponding function: y ⴝ cos x ⴙ 2 cos2x ⴚ 2
Determine the approximate zeros in the given domain.
The roots are approximately: x ⴝ — 0.67 and x ⴝ — 5.61
Substitute each root into the given equation to verify.
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Chapter 7: Trigonometric Equations and Identities—Checkpoint—Solutions
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3. Use graphing technology to determine the general solution of each
equation over the set of real numbers. Give the answers to the
nearest hundredth.
a) 4 tan x ⫺ 5 ⫽ 0
Graph the corresponding function: y ⴝ 4 tan x ⴚ 5
The period of the function is ␲.
Determine the zero in the domain 0 ◊ x<␲.
The root is approximately: x ⴝ 0.90
The general solution is approximately: x ⴝ 0.90 ⴙ ␲k, k ç ⺪
b) 6 cos2x ⫹ cos x ⫽ 1
Graph the corresponding function: y ⴝ 6 cos2x ⴙ cos x ⴚ 1
The period of the function is 2␲.
Determine the zeros in the domain 0 ◊ x<2␲.
The roots are approximately: x ⴝ 1.23, x ⴝ 2.09, x ⴝ 4.19, x ⴝ 5.05
The general solution is approximately: x ⴝ 1.23 ⴙ 2␲k, k ç ⺪ or
x ⴝ 2.09 ⴙ 2␲k, k ç ⺪ or x ⴝ 4.19 ⴙ 2␲k, k ç ⺪ or x ⴝ 5.05 ⴙ 2␲k,
kç⺪
7.2
4. Multiple Choice Which number is a root of the equation
3 sin x ⫹ 1 ⫽ 5 sin x ⫺ 1 over the domain 0 ≤ x < 2␲?
A. 0
␲
B. ␲
3␲
C. 2
D. 2
√
5. Use algebra to solve the equation 2 cos 2x ⫹ 1 ⫽ 0 over the
domain ⫺␲ < x < ␲, then write the general solution of the
equation.
√
2 cos 2x ⴝ ⴚ 1
1
2
cos 2x ⴝ ⴚ√
The terminal arm of angle 2x lies in Quadrant 2 or 3.
The reference angle for angle 2x is: cosⴚ1 a√ b ⴝ
1
2
␲
4
3␲
4
3␲
x ⴝⴚ
8
3␲
4
3␲
xⴝ
8
In Quadrant 2, 2x ⴝ
In Quadrant 3, 2x ⴝ ⴚ
The period of cos 2x is ␲, so other roots are:
3␲
ⴙ␲
8
5␲
xⴝ
8
3␲
5␲
The roots are: x ⴝ —
and x ⴝ —
8
8
3␲
5␲
The general solution is: x ⴝ
ⴙ ␲k, k ç ⺪ or x ⴝ
ⴙ ␲k, k ç ⺪
8
8
3␲
ⴚ␲
8
5␲
xⴝ ⴚ
8
xⴝ
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and
x ⴝⴚ
Chapter 7: Trigonometric Equations and Identities—Checkpoint—Solutions
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␲
5␲
6. Verify that 6 and 6 are two roots of the equation 4 cos2x ⫺ 3 ⫽ 0.
Substitute each given value in the equation.
For x ⴝ
␲
:
6
For x ⴝ
␲
6
L.S. ⴝ 4 cos2 a b ⴚ 3
√
ⴝ 4a
5␲
:
6
L.S. ⴝ 4 cos2 a
2
3
b ⴚ3
2
ⴝ 4 aⴚ
ⴝ0
ⴝ R.S.
√
5␲
b –3
6
2
3
b ⴚ3
2
ⴝ0
ⴝ R.S.
For each value of x, the left side is equal to the right side, so the roots
are verified.
7. Use algebra to solve the equation 10 sin2x ⫹ 11 sin x ⫽ ⫺3 over the
domain 90° ≤ x ≤ 360°. Give the roots to the nearest degree.
10 sin2x ⴙ 11 sin x ⴙ 3 ⴝ 0
(2 sin x + 1)(5 sin x + 3) ⴝ 0
or 5 sin x ⴙ 3 ⴝ 0
Either 2 sin x ⴙ 1 ⴝ 0
sin x ⴝ ⴚ0.5
ⴚ1
sin x ⴝ ⴚ0.6
The reference angle is: sin (0.5) ⴝ 30°
The reference angle is: sinⴚ1(0.6) ⬟ 37°
The terminal arm of angle x lies in
Quadrant 3 or 4.
The terminal arm of angle x lies in
Quadrant 3 or 4.
In Quadrant 3, x ⴝ 180° ⴙ 30°, or 210°
In Quadrant 3, x ⬟ 180° + 37°, or 217°
In Quadrant 4, x ⴝ 360° ⴚ 30°, or 330°
In Quadrant 4, x ⬟ 360° ⴚ 37°, or 323°
The roots are: x ⴝ 210°, x ⬟ 217°, x ⬟ 323°, x ⴝ 330°
16
Chapter 7: Trigonometric Equations and Identities—Checkpoint—Solutions
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