1© 2011 Pearson Education, Inc. All rights reserved 1© 2010 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved

Chapter 6

Trigonometric Identities and

Equations

OBJECTIVES

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Double-Angle and Half-Angle IdentitiesSECTION 6.3

1

2

Use double-angle identities.Use power-reducing identities.Use half-angle identities.3

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DOUBLE-ANGLE IDENTITIES

1cos22cos

sin212costan1

tan22tan

sincos2coscossin22sin

2

22

22

xx

xxx

xx

xxxxxx

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EXAMPLE 1 Using Double-Angle Identities

If and is in quadrant II, find the

exact value of each expression.

cos 3

5

Solution

a. sin 2 b. cos2 c. tan 2

Use identities to find sin θ and tan θ.

2 9 4sin 1 cos 1

25 5 θ is in QII so

sin > 0.sin 4 / 5 4

tancos 3 / 5 3

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EXAMPLE 1

Solution continued

a cos. sin 2 2sin

Using Double-Angle Identities

24

25

52

4

5

3

2 2cosb. s cos2 in

9

25

16

25

2 2

5

3

5

4

7

25

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EXAMPLE 1

Solution continued

2

ta2c.

n

ttan

n2

1 a

Using Double-Angle Identities

83

1 169

2

2

1

4

3

4

3

83

79

8

3

9

7

24

7

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EXAMPLE 3

Solution

Finding a Triple-Angle Identity for Sines

Verify the identity sin 3x = 3 sin x – 4 sin3 x.

sin 3x = sin (2x + x)

= sin 2x cos x + cos 2x sin x

= (2 sin x cos x) cos x + (1 – 2 sin2 x) sin x

= 2 sin x cos2 x + sin x – 2 sin3 x

= 2 sin x (1 – sin2 x) + sin x – 2 sin3 x

= 2 sin x – 2 sin3 x + sin x – 2 sin3 x

= 3 sin x – 4 sin3 x

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POWER REDUCING IDENTITIES

sin2 x 1 cos2x

2

cos2 x 1 cos2x

2

tan2 x 1 cos2x

1 cos2x

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EXAMPLE 4 Using Power-Reducing Identities

Write an equivalent expression for cos4 x that

contains only first powers of cosines of multiple angles.Solution

Use power-reducing identities repeatedly.

24 2cos cosx x

1 cos2x

2

2

211 2cos2 cos 2

4x x

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EXAMPLE 4

Solution continued

Using Power-Reducing Identities

1

41 2 cos2x

1 cos 4x

2

1

41 2 cos2x

1

2

1

2cos 4x

1

4

2

4cos2x

1

8

1

8cos 4x

3

8

1

2cos2x

1

8cos 4x

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HALF-ANGLE IDENTITIES

1 cossin

2 2

1 coscos

2 2

1 costan

2 1 cos

The sign, + or –, depends on the quadrant in

which lies.2

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EXAMPLE 6 Using Half-Angle Identities

Use a half-angle formula to find the exact value of cos 157.5º.

Solution

cos157.5º cos315º

2

1 cos315º

2

Because 157.5º = , use the half-angle identity

for cos with θ = 315°. Because

lies in quadrant II, cos is negative.

2

315

2

5.1572

2

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EXAMPLE 6

Solution continued

cos 5

2

º1 4

2

21 2

2 2

2 2

2 2

2 2

2

Using Half-Angle Identities