This document provides examples of solving trigonometric equations. It shows how to determine all solutions by considering the periodic nature of trigonometric functions like sine, cosine, and tangent. Example problems include finding solutions to equations like 2sin^2 x = 3, cos^2 θ = -1/3, and tan 3x = -1 within a given interval by analyzing the quadrant(s) where the trig function is positive or negative. Solutions may involve adding integer multiples of the period (2π for sine/cosine, π for tangent) to account for all possibilities.

1. MATH 107
Section 6.5
Trigonometric Equations

2. Determine whether is a solution of the equation 2sin 2 0.
4
5
Is a solution?
4
π
θ θ
π
θ
= + =
=

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EXAMPLE 1 Solving a Trigonometric Equation
Find all solutions of each equation. Express all
solutions in radians.
2
a. sin
2
x =
3
b. cos
2
θ = −
c. tan 3x = −

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a. First find all solutions in [0, 2π).
We know and sin x > 0 in quadrants
I and II.
QI and QII angles with reference angles of
are and .
EXAMPLE 1 Solving a Trigonometric Equation
Solution
2
a. sin
2
x =

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Since sin x has a period of 2π, all solutions of the
equation are given by
or
for any integer n.
EXAMPLE 1 Solving a Trigonometric Equation
Solution continued

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EXAMPLE 1 Solving a Trigonometric Equation
Solution
a. First find all solutions in [0, 2π).
We know and cos θ < 0 in
quadrants II and III.
QII and QIII angles with reference angles of
are and .
3
b. cos
2
θ = −
3
cos
6 2
π
=
6
π
5
6 6
π π
θ π= − =
7
6 6
π π
θ π= + =

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EXAMPLE 1 Solving a Trigonometric Equation
Solution continued
Since cos θ has a period of 2π, all solutions of
the equation are given by
or
for any integer n.
5
2
6
n
π
θ π= +
7
2
6
n
π
θ π= +

8. 8© 2011 Pearson Education, Inc. All rights reserved
The QII angle with a reference angle of is
.
We know and tan x < 0 in
quadrant II.
EXAMPLE 1 Solving a Trigonometric Equation
Solution
a. Because tan x has a period of π, first find all
solutions in [0, π).
tan 3
3
π
=
3
π
c. tan 3x = −
3
2
3
ππ
π =−=x

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EXAMPLE 1 Solving a Trigonometric Equation
Solution continued
Since tan x has a period of π, all solutions of the
equation are given by
for any integer n.
π
π
nx +=
3
2

10. 2cos 3 0θ − =
(Answers on next slide.)

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The reference angle is
because
In QI and QII, sin θ > 0.
EXAMPLE 3 Solving a Linear Trigonometric Equation
Find all solutions in the interval [0, 2π) of the
equation .2sin 1 2
4
x
π 
− + = ÷
 
2sinθ +1 = 2
2sinθ = 1
sinθ =
1
2
Solution
Replace with θ in the given equation.
4
π
−x
6
π
.
2
1
6
sin =
π

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EXAMPLE 3
Solution continued
Solving a Linear Trigonometric Equation
6
θ
π
=
5
6
θ
π
=or
64
x
ππ
− =
x =
π
6
+
π
4
x =
2π
12
+
3π
12
=
5π
12
4
5
6
x
ππ
− =
x =
5π
6
+
π
4
x =
10π
12
+
3π
12
=
13π
12
or
.
12
13
,
12
5





 ππ
The solution set in the interval [0, 2π) is

13. Solve the equation: 2cos 2θ( ) −1= 0
(Answers on next slide.)

14. Solve the equation: cos θ −
π
4





÷=1
(Answers on next slide.)

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EXAMPLE 6 Solving a Quadratic Trigonometric Equation
Find all solutions of the equation
2sin2
θ − 5sinθ + 2 = 0.
Express the solutions in radians.
Solution
Factor 2sin2
θ − 5sinθ + 2 = 0.
( ) ( )2sin 1 sin 2 0θ θ− − =
( ) ( )2sin 1 0 or sin 2 0θ θ− = − =
sinθ =
1
2
sinθ = 2
θ =
π
6
or θ =
5π
6
No solution

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π
π
θπ
π
θ nn 2
6
5
or2
6
+=+=
EXAMPLE 6
Solution continued
So, θ =
π
6
and θ =
5π
6
Since sinθ has a period of 2π, the solutions are
for any integer n.
are the only solutions
in the interval [0, 2π).
Solving a Quadratic Trigonometric Equation

17. Solving Trigonometric Equations
2
2sin sin 1 0x x− − =Solve: