7.4 Double Angle and Half Angle Identities

7-4 NAME _____________________________ DATE _______________ PERIOD ________

Study Guide

Double-Angle and Half-Angle Identities

Example 1 If sin ␪ ϭ ᎏ41ᎏ and ␪ has its terminal side in the first
quadrant, find the exact value of sin 2␪.

To use the double-angle identity for sin 2␪, we must
first find cos ␪.

sin2 ␪ ϩ cos2 ␪ ϭ 1 sin ␪ ϭ ᎏ41ᎏ

΂ ΃ᎏ41ᎏ 2 ϩ cos2 ␪ ϭ 1

cos2 ␪ ϭ ᎏ115ᎏ6
cos ␪ ϭ ᎏ͙41ෆᎏ5ෆ

Now find sin 2␪.

sin 2␪ ϭ 2 sin ␪ cos ␪ Double-angle identity for sine
sin ␪ ϭ ᎏ14ᎏ, cos ␪ ϭ ᎏ͙4ෆ1ᎏෆ5
΂ ΃ϭ 2 ᎏ41ᎏ ᎏ͙41ෆᎏ5ෆ

ϭ ᎏ͙81ෆᎏෆ5

Example 2 Use a half-angle identity to find the exact value
of sin ᎏ1␲ᎏ2 .
sin ᎏ1␲2ᎏ ϭ sin ᎏᎏ␲26ᎏ

Ί๶ϭ ᎏ1 Ϫ c2ᎏos ᎏ␲6ᎏ Ί๶๶๶๶๶๶Use sin ᎏ␣2ᎏ ϭ Ϯ ᎏ1 Ϫ 2cᎏos a. Since ᎏ1␲2ᎏ is in

Quadrant I, choose the positive sine value.

Ί๶ϭ ᎏ1Ϫ2ᎏ͙2ᎏෆ3

Ί๶๶๶๶ϭ ᎏ2 Ϫ4ᎏ͙ෆ3

ϭ ᎏ͙ෆ2ෆϪ2ᎏෆ͙3ෆෆ

© Glencoe/McGraw-Hill 284 Advanced Mathematical Concepts

7-4 NAME _____________________________ DATE _______________ PERIOD ________

Practice

Double-Angle and Half-Angle Identities

Use a half-angle identity to find the exact value of each function.

1. sin 105؇ 2. tan ᎏ␲8ᎏ 3. cos ᎏ58␲ᎏ

Use the given information to find sin 2␪, cos 2␪, and tan 2␪.

4. sin ␪ ϭ ᎏ1123ᎏ, 0؇ Ͻ ␪ Ͻ 90؇ 5. tan ␪ ϭ ᎏ12ᎏ, ␲ Ͻ ␪ Ͻ ᎏ32␲ᎏ

6. sec ␪ ϭ Ϫᎏ25ᎏ, ᎏ␲2ᎏ Ͻ ␪ Ͻ ␲ 7. sin ␪ ϭ ᎏ53ᎏ, 0 Ͻ ␪ Ͻ ᎏ␲2ᎏ

Verify that each equation is an identity.
8. 1 ϩ sin 2x ϭ (sin x ϩ cos x)2

9. cos x sin x ϭ ᎏsin2ᎏ2x

10. Baseball A batter hits a ball with an initial velocity v0 of
100 feet per second at an angle ␪ to the horizontal. An outfielder
catches the ball 200 feet from home plate. Find ␪ if the range
of a projectile is given by the formula R ϭ ᎏ31ᎏ2 v02 sin 2␪.

© Glencoe/McGraw-Hill 285 Advanced Mathematical Concepts