Reteach 14-6

Name Date Class
LESSON
14-6
Reteach
Solving Trigonometric Equations
You can use the same methods to solve trigonometric equations as to solve algebraic
equations.
Substitute variables to solve trigonometric equations that resemble quadratic equations.
Solve: si n 2 3sin 4 0 for 0 360
Step 1 Make a substitution.
Let x sin in si n 2 3sin 4 0
x 2 3x 4 0
Step 2 Solve the quadratic x 2 3x 4 0.
x 4 x 1 0
Factor.
x 4 0 x 1 0 Set each factor equal to 0.
x 4 x 1 Solve each equation for x.
Step 3 Substitute sin for x.
sin 4
–1 sin 1
This equation has no solution.
sin 1 Think: What angle has a sine of 1?
270
The only solution is 270.
Complete to solve each equation for 0 360.
1. 3t an 2 1 0 2. 2s in 2 sin 0
Let x tan . Let x sin .
Hint: There are 4 solutions.
Hint: There are 4 solutions.
3x 2 1 0 2 x 2 x 0
3x 2 1 x 2x 1 0
x ___ 3
x 0 or 2x 1 0
3
tan ___ 3
3 or ___ 3
3 ; x 0 or x __ 1 2
sin 0 or sin __ 1
30, 150, 210, 330
0, 30, 150, 180 2
3. cos 2 2cos 1 0 4. 2c os 2 7cos 3 0
x 2 2x 1; x 1 2 ;
cos 1; 0
The interval includes 0 but
not 360.
2 x 2 7x 3 0
2x 1 x 3 0
x __ 1 2 or x 3
60, 300
Copyright © by Holt, Rinehart and Winston.
46 Holt Algebra 2
All rights reserved.

Name Date Class
LESSON
14-6
Reteach
Solving Trigonometric Equations (continued)
You can use trigonometric identities to solve trigonometric equations. This
is especially useful when more than one trigonometric function appears in
an equation.
Use an identity to rewrite the equation using a single
trigonometric function or a single angle.
Solve: cos2 cos 0 for 0 360
Step 1 The angles are not the same.
Use the double-angle identity for cosine.
Choose cos2 2c os 2 1.
Use an identity. Choose the
formula that allows you to
write the equation in terms of
cos only.
Step 2 Substitute 2c os 2 1 for cos2 in the original equation.
cos2 cos 0
Write the original equation.
2co s 2 1 cos 0
Substitute.
2c os 2 cos 1 0
Write in standard form.
Step 3 Solve the equation.
2c os 2 cos 1 0
2cos 1 cos 1 0
Factor.
2cos 1 0 cos 1 0 Set each factor equal to 0.
cos 1__ 2
cos 1 Solve each equation for cos .
Step 4 Find the values that satisfy each solution.
cos 1__ Think: What angle has a cosine of 1__ 2
2 ?
60, 300
cos 1 Think: What angle has a cosine of 1?
180
Remember to
find all values
in the interval.
Use trigonometric identities to solve each equation for 0 360.
5. cos2 sin 2 0 6. c os 2 2sin 1
Use cos2 1 2 sin 2 . Use c os 2 1 s in 2 .
1 2si n 2 sin 2 0
2s in 2 – sin 3 0
2sin 3 sin 1 0
sin __ 3 or sin 1
2
270
1 s in 2 2sin 1
si n 2 2sin 0
sin sin 2 0
sin 0 or sin 2
0, 180
Copyright © by Holt, Rinehart and Winston.
47 Holt Algebra 2
All rights reserved.