SECTION 6 3 DoubleAngle and HalfAngle Identities OBJECTIVES

SECTION 6. 3 Double-Angle and Half-Angle Identities OBJECTIVES 1 2 3 Use double-angle identities. Use power-reducing identities. Use half-angle identities.

DOUBLE-ANGLE IDENTITIES © 2011 Pearson Education, Inc. All rights reserved 2

© 2011 Pearson Education, Inc. All rights reserved 3

EXAMPLE 1 Using Double-Angle Identities If and is in quadrant II, find the exact value of each expression. Solution First, we use identities to find sin θ and tan θ. θ is in QII so sin > 0. © 2011 Pearson Education, Inc. All rights reserved 4

EXAMPLE 1 Using Double-Angle Identities Solution continued © 2011 Pearson Education, Inc. All rights reserved 5

EXAMPLE 1 Using Double-Angle Identities Solution continued © 2011 Pearson Education, Inc. All rights reserved 6

Using the Double-Angle Formula for Tangent to Find an Exact Value © 2011 Pearson Education, Inc. All rights reserved 7

© 2011 Pearson Education, Inc. All rights reserved 8

Verifying an Identity © 2011 Pearson Education, Inc. All rights reserved 9

Verifying an Identity (continued) © 2011 Pearson Education, Inc. All rights reserved 10

EXAMPLE 3 Finding a Triple-Angle Identity for Sines Verify the identity sin 3 x = 3 sin x – 4 sin 3 x. Solution sin 3 x = sin (2 x + x) = sin 2 x cos x + cos 2 x sin x = (2 sin x cos x) cos x + (1 – 2 sin 2 x) sin x = 2 sin x cos 2 x + sin x – 2 sin 3 x = 2 sin x (1 – sin 2 x) + sin x – 2 sin 3 x = 2 sin x – 2 sin 3 x + sin x – 2 sin 3 x = 3 sin x – 4 sin 3 x © 2011 Pearson Education, Inc. All rights reserved 11

POWER REDUCING IDENTITIES © 2011 Pearson Education, Inc. All rights reserved 12

EXAMPLE 4 Using Power-Reducing Identities Write an equivalent expression for cos 4 x that contains only first powers of cosines of multiple angles. Solution Use power-reducing identities repeatedly. © 2011 Pearson Education, Inc. All rights reserved 13

EXAMPLE 4 Using Power-Reducing Identities Solution continued © 2011 Pearson Education, Inc. All rights reserved 14

HALF-ANGLE IDENTITIES The sign, + or –, depends on the quadrant in which lies. © 2011 Pearson Education, Inc. All rights reserved 15

EXAMPLE 6 Using Half-Angle Identities Use a half-angle formula to find the exact value of cos 157. 5º. Solution Because 157. 5º = , use the half-angle identity for cos with θ = 315°. Because lies in quadrant II, cos is negative. © 2011 Pearson Education, Inc. All rights reserved 16

EXAMPLE 6 Using Half-Angle Identities Solution continued © 2011 Pearson Education, Inc. All rights reserved 17

Verifying an Identity 18

Half-Angle Formulas for Tangent © 2011 Pearson Education, Inc. All rights reserved 19

Verifying an Identity We worked with the right side and arrived at the left side. Thus, the identity is verified. © 2011 Pearson Education, Inc. All rights reserved 20

© 2011 Pearson Education, Inc. All rights reserved 21