Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from...
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![Page 1: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/1.jpg)
Solve by Isolating Trigonometric Expressions
Solve .
Original equation
Subtract 3cos x from each side to isolate the trigonometric expression.
Solve for cos x.
![Page 2: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/2.jpg)
Solve by Isolating Trigonometric Expressions
Answer:
The period of cosine is 2π, so you only need to find
solutions on the interval [0, 2π). The solutions on this
interval are . The solutions on the interval
(–∞, ∞) are then found by adding integer multiples 2π.
Therefore, the general form of the solutions is
x = , where n is an integer.
![Page 3: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/3.jpg)
Solve sin x + = – sin x.
A.
B.
C.
D.
![Page 4: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/4.jpg)
Solve by Taking the Square Root of Each Side
Solve 3 tan2 x – 4 = –3.
3 tan2 x – 4 = –3 Original equation
3 tan2 x = 1 Add 4 to each side.
Divide each side by 3.
Take the square root of each side.
Rationalize the denominator.
![Page 5: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/5.jpg)
Solve by Taking the Square Root of Each Side
The period of tangent is π. On the interval [0, π),
tan x = when x = and tan x = when x = .
The solutions on the interval (–∞, ∞) have the general
form , where n is an integer.
Answer:
![Page 6: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/6.jpg)
Solve 5 tan2x – 15 = 0.
A.
B.
C.
D.
![Page 7: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/7.jpg)
Solve by Factoring
A. Find all solutions of on the interval [0, 2π).
Original equation
Isolate the trigonometric terms.
Factor.
Zero Product Property
![Page 8: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/8.jpg)
Solve by Factoring
Solve for x on [0, 2π).
On the interval [0, 2π), the equation
has solutions .
Answer:
Solve for cos x.
![Page 9: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/9.jpg)
Solve by Factoring
B. Find all solutions of 2sin2x + sinx – 1 = 0 on the interval [0, 2π).
Original equation
Factor.
Zero Product PropertySolve for sin x.
Solve for x on [0, 2π).
![Page 10: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/10.jpg)
Solve by Factoring
Answer:
On the interval [0, 2π), the equation 2sin2x + sinx – 1 = 0
has solutions .
![Page 11: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/11.jpg)
Find all solutions of 2 tan4 x – tan2 x – 15 = 0 on the interval [0, π).
A.
B.
C.
D.
![Page 12: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/12.jpg)
Trigonometric Functions of Multiple Angles
PROJECTILES A projectile is sent off with an initial
speed vo of 350 m/s and clears a fence 3000 m
away. The height of the fence is the same height as
the initial height of the projectile. If the distance the
projectile traveled is given by , find the
interval of possible launch angles to clear the
fence.
![Page 13: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/13.jpg)
Trigonometric Functions of Multiple Angles
Original formula
d = 3000 and v0 = 350
Simplify.
Multiply each side by 9.8.
Divide each side by 122,500.
Definition of inverse sine.
![Page 14: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/14.jpg)
Trigonometric Functions of Multiple Angles
Recall from Lesson 4-6 that the range of the inverse sine function is restricted to acute angles of θ in the interval [–90°, 90°]. Since we are finding the inverse sine of 2θ instead of θ, we need to consider angles in the interval [–2(90°), 2(90°)] or [–180°, 180°]. Use your calculator to find the acute angle and the reference angle relationship sin (180° − θ) = sin θ to find the obtuse angle.
sin–10.24 = 2 Definition of inverse sine
13.9° or 166.1°= 2sin–1(0.24) ≈13.9° and sin(180° – 13.9°) = 166.1°
7.0° or 83.1° = Divide by 2.
![Page 15: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/15.jpg)
Answer: 7.0° ≤ ≤ 83.1°
Trigonometric Functions of Multiple Angles
The interval is [7.0°, 83.1°]. The ball will clear the fence if the angle is between 7.0° and 83.1°.
CHECK Substitute the angle measures into the original equation to confirm the solution.
Original formula
Use a calculator.
= 7.0° or = 83.1°
![Page 16: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/16.jpg)
GOLF A golf ball is sent off with an initial speed vo
of 36 m/s and clears a small barricade 70 m away.
The height of the barricade is the same height as
the initial height of the ball. If the distance the ball
traveled is given by , find the interval
of possible launch angles to clear the barricade.
A. 1.6° ≤ ≤ 88.5°
B. 3.1° ≤ ≤ 176.9°
C. 16.0° ≤ ≤ 74.0°
D. 32° ≤ ≤ 148.0°
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Solve by Rewriting Using a Single Trigonometric Function
Find all solutions of sin2 x – sin x + 1 = cos2 x on the interval [0, 2π).
sin2 x – sin x + 1
= cos2 x
Original equation
–cos2 x + sin2 x – sin x + 1
= 0
Subtract cos2 x from each side.
–(1 – sin2 x) + sin2 x – sin x + 1
= 0
Pythagorean Identity
2sin2 x – sin x
= 0
Simplify.
sin x (2sin x – 1)
= 0
Factor.
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Solve by Rewriting Using a Single Trigonometric Function
sin x = 02sin x – 1= 0Zero Product Property
2sin x = 1Solve for sin x.
Solve for x on [0, 2π).
x = 0, π
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Solve by Rewriting Using a Single Trigonometric Function
CHECK The graphs of Y1 = sin2 x – sin x + 1 and
Y2 = cos2 x intersect at on the interval
[0, 2π) as shown.
Answer:
![Page 20: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/20.jpg)
Find all solutions of 2sin2x = cosx + 1 on the interval [0, 2).
A.
B.
C.
D.
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Find all solutions of sin x – cos x = 1 on the interval [0, 2π).
sin x – cos x= 1Original equation
sin x= cos x + 1Add cos x to each side.
sin2 x= cos2 x + 2cos x + 1 Square each side.
1 – cos2 x= cos2 x + 2cos x + 1Pythagorean Identity
0= 2cos2 x + 2cos xSubtract 1 – cos2x from each side.
0= cos2 x + cos x Divide each side by 2.
0= cos x(cos x + 1)Factor.
Solve by Squaring
![Page 22: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/22.jpg)
Solve by Squaring
cos x = 0 cos x + 1= 0 Zero Product Property
cos x= –1 Solve for cos x.
Original formula
Simplify.
, x = πSolve for x on [0, 2).
Substitute sin π – cos π = 1
![Page 23: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/23.jpg)
Solve by Squaring
Therefore, the only valid solutions are on the interval .
Answer:
![Page 24: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/24.jpg)
Find all solutions of 1 + cos x = sin x on the interval [0, 2π).
A.
B.
C.
D.
![Page 25: Example 1 Solve by Isolating Trigonometric Expressions Solve. Original equation Subtract 3cos x from each side to isolate the trigonometric expression.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649e8f5503460f94b93c1e/html5/thumbnails/25.jpg)