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

3 © 2011 Pearson Education, Inc. All rights reserved DOUBLE-ANGLE IDENTITIES

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

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

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

7 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 3 Solution Finding a Triple-Angle Identity for Sines Verify the identity sin 3x = 3 sin x – 4 sin 3 x. sin 3x = sin (2x + x) = sin 2x cos x + cos 2x 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

8 © 2011 Pearson Education, Inc. All rights reserved POWER REDUCING IDENTITIES

9 © 2011 Pearson Education, Inc. All rights reserved 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.

10 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 4 Solution continued Using Power-Reducing Identities

11 © 2011 Pearson Education, Inc. All rights reserved HALF-ANGLE IDENTITIES The sign, + or –, depends on the quadrant in which lies.

12 © 2011 Pearson Education, Inc. All rights reserved 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.

13 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 6 Solution continued Using Half-Angle Identities