MATH 107

Section 6.5

Trigonometric Equations

Determine whether is a solution of the equation 2sin 2 0. 4

5Is a solution?

4

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EXAMPLE 1 Solving a Trigonometric Equation

Find all solutions of each equation. Express all solutions in radians.

2a. sin

2x

3b. cos

2

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

2a. sin

2x

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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 .

3b. cos

2

3cos

6 2

6

5

6 6

7

6 6

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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.

52

6n

72

6n

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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 33

3

c. tan 3x

3

2

3

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.

nx

3

2

2cos 3 0

(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 24

x

Solution

Replace with θ in the given equation.4

x

6

.2

1

6sin

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EXAMPLE 3

Solution continued

Solving a Linear Trigonometric Equation

6

5

6

or

64x

4

5

6x

or

.12

13,

12

5

The solution set in the interval [0, 2π) is

(Answers on next slide.)

(Answers on next slide.)

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EXAMPLE 6 Solving a Quadratic Trigonometric Equation

Find all solutions of the equation

Express the solutions in radians.Solution

Factor 2sin 1 sin 2 0

2sin 1 0 or sin 2 0

No solution

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nn 26

5or 2

6

EXAMPLE 6

Solution continued

So,

Since sin 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

Solving Trigonometric Equations22sin sin 1 0x x Solve: