Slide 6.2 - 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Trigonometric Equations

Learn to solve trigonometric equations of the form a sin (x – c) = k, a cos (x – c) = k, and a tan (x – c) = k.Learn to solve trigonometric equations involving multiple angles.Learn to solve trigonometric equations by using the zero-product property. Learn to solve trigonometric equations that involve more than one trigonometric function.Learn to solve trigonometric equations by squaring both sides.

SECTION 6.2

1

2

3

4

5

Slide 6.2 - 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

TRIGONOMETRIC EQUATIONSA trigonometric equation is an equation that contains a trigonometric function with a variable.

Equations that are true for all values in the domain of the variable are called identities.

Equations that are true for some but not all values of the variable are called conditional equations.Solving a trigonometric equation means to find its solution set.

Slide 6.2 - 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 3 Solving a Linear Trigonometric Equation

Find all solutions in the interval [0, 2π) of the

equation: 2sin x 4

1 2

Solution

Replace by in the given equation.x 4

2sin 1 2

2sin 1

sin 1

2

sin6

1

2We know

sin is (+) in Q I and II

6

, 56

Slide 6.2 - 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 3

Solution continued

Solving a Linear Trigonometric Equation

x 4

6

56

and or

x 4

6

x 6

4

x 212

312

512

x 4

56

x 56

4

x 1012

312

1312

512

,1312

.Solution set in [0, 2π] is

Slide 6.2 - 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 4Solving a Trigonometric Equation Containing Multiple Angles

Find all solutions in the interval [0, 2π) of the

equation: cos 3x 1

2

Period of cosine function is 2π. Replace with 3x.

cos is (+) in Q I and IV,

Solution

Recall cos3

1

2.

3x 3

2n 3x 53

2nSo or

3

, 53

so

Slide 6.2 - 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 3

Solution continued

Solving a Linear Trigonometric Equation

x 9

2n

3x

59

2n

3Or or

To find solutions in the interval [0, 2π), try:

x 9

23

59

x 59

23

9

x 9

x 59

x 9

23

79

x 59

23

11

9

n = –1

n = 0

n = 1

Slide 6.2 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 3

Solution continued

Solving a Linear Trigonometric Equation

9

,59

,79

,11

9,13

9,17

9

.Solution set is

Values resulting from n = –1 are too small.

x 9

43

13

9x

59

43

17

9

x 9

2 19

9x

59

2 23

9

n = 2

n = 3

Values resulting from n = 3 are too large.Solutions we want correspond to n = 0, 1, and 2.

Slide 6.2 - 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 7 Solving a Quadratic Trigonometric Equation

Find all solutions of the equation

2sin2 5sin 2 0.Express the solutions in radians.Solution

Factor 2sin2 5sin 2 0.2sin 1 sin 2 0

2sin 1 0 or sin 2 0

sin 1

2sin 2

6

or 56

No solution because –1 ≤ sin ≤ 1.

Slide 6.2 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 7

Solution continued

So, 6

and 56

Since sin has period 2π, the solutions of the given equation are

6

2n or 56

2n ,

where n is any integer.

are the only two

solutions in the interval [0, 2π).

Solving a Quadratic Trigonometric Equation

Slide 6.2 - 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 8Solving a Trigonometric Equation Using Identities

Find all the solutions in the interval [0, 2π) to the equation 2sin2 3 cos 1 0.

Solution

Use the Pythagorean identity to rewrite the equation in terms of cosine only.

2sin2 3 cos 1 0

2 1 cos2 3 cos 1 0

2 2 cos2 3 cos 1 0

3 2 cos2 3 cos 0

Slide 6.2 - 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 8

Solution continued

2 cos2 3 cos 3 0

Solving a Trigonometric Equation Using Identities

Use the quadratic formula to solve this equation.

cos 3 3 2 4 2 3

2 2

cos 3 3 24

4

3 27

4

3 3 3

4

Slide 6.2 - 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 8

Solution continued

Solving a Trigonometric Equation Using Identities

Hence,

cos 3 3 3

4 or cos

3 3 3

4

cos 4 3

4

cos 3 1

No solution because –1 ≤ cos ≤ 1.

cos 2 3

4

cos 3

2

cos is (–) in QII, QIII

Slide 6.2 - 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 8

Solution continued

Solving a Trigonometric Equation Using Identities

cos 3

2 when

6

6

56

6

76

Solution set in the interval [0, 2π) is56

,76

.

Slide 6.2 - 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 9Solving a Trigonometric Equation by Squaring

Solution

Square both sides and use identities to convert to an equation containing only sin x.

Find all the solutions in the interval [0, 2π) to the equation 3 cos x sin x 1.

3 cos x sin x 1.

3 cos x 2 sin x 1 2

3cos2 x sin2 x 2sin x 1

3 1 sin2 x sin2 x 2sin x 1

Slide 6.2 - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 9

Solution continued

3 3sin2 x sin2 x 2sin x 1

Solving a Trigonometric Equation by Squaring

4 sin2 x 2sin x 2 0

2sin2 x sin x 1 02sin x 1 sin x 1 0

2sin x 1 0 or sin x 1 0

sin x 1

2

x 6

or 56

sin x 1

x 32

Slide 6.2 - 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 9

Solution continuedPossible solutions are:

Solving a Trigonometric Equation by Squaring

x 6

x 32

x 56

3 cos6

?

sin6

1

Solution set in the interval [0, 2π) is6

,32

.

3

2

3

2

3 cos56

?

sin56

1 3

2? 3

2

3 cos32

?

sin32

1 0 0