Double-Angle and Half-Angle Identities SECTION 6.3 Use double-angle identities. Use power-reducing identities. Use half-angle identities. 1 2 3

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

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

© 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 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

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

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

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

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

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

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

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 3 Finding a Triple-Angle Identity for Sines Verify the identity sin 3x = 3 sin x – 4 sin3 x. Solution 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 © 2011 Pearson Education, Inc. All rights reserved

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

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

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

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

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

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

Verifying an Identity  

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

© 2011 Pearson Education, Inc. All rights reserved 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

© 2011 Pearson Education, Inc. All rights reserved