Double-Angle and Power-Reducing

IdentitiesSection 5.4

Double-Angle Identitiessin 2 2sin cosu u u

2 2

2

2

cos sin

cos 2 2cos 1

1 2sin

u u

u u

u

2

2 tantan 21 tan

uuu

Proving the first of these:

sin 2u sin u u sin cos cos sinu u u u 2sin cosu u

Power-Reducing Identities2 1 cos 2sin

2uu

2 1 cos 2cos2uu

2 1 cos 2tan1 cos 2

uuu

Guided Practice2cos2 2cos 1u u Prove the given identity.

cos 2u cos u u cos cos sin sinu u u u

2 2cos sinu u

2 2cos 1 cosu u 22cos 1u

Guided Practice2

2 tantan 21 tan

uuu

Prove the given identity.

tan 2u tan u u tan tan1 tan tan

u uu u

2

2 tan1 tan

uu

Guided Practice4 4cos sin cos 2 Prove the given identity.

2 2 2 2cos sin cos sin 4 4cos sin

2 21 cos sin

cos 2

Guided Practice4cos xRewrite in terms of trigonometric functions with

no power greater than 1.

22cos x4cos x21 cos2

2x

21 2cos2 cos 2

4x x

1 1 1 1 cos 4cos 24 2 4 2

xx

1 1 1 1cos 2 cos 44 2 8 8

x x 1 3 4cos 2 cos 48

x x

Guided Practice 0,2Find all solutions to the given equation in the interval .

sin 2 sinx xsin 2 sin 0x x

2sin cos sin 0x x x sin 2cos 1 0x x

sin 0x or1cos2

x

50, , ,3 3

x

Guided Practice 0,2Find all solutions to the given equation in the interval .

cos 2 cosx xcos2 cos 0x x

22cos 1 cos 0x x 22cos cos 1 0x x

1cos2

x or cos 1x

2 40, ,3 3

x

2cos 1 cos 1 0x x

Guided Practice 0,2Find all solutions to the given equation in the interval .

2cos cos cos 2x x x 2 2cos cos 2cos 1x x x

20 cos cos 1x x

1 5cos2

x

Quadratic Formula: Only keep this answer:

1 5cos2

x

Guided Practice 0,2Find all solutions to the given equation in the interval .

2cos cos cos 2x x x

or

1 5cos2

x

1 1 5cos 2.2372

x

1 1 52 cos 4.046

2x

Half-Angle Identities

Half-Angle Identities

1 cossin2 2u u

1 coscos2 2u u

1 cos1 cos

1 costan2 sin

sin1 cos

uu

u uuuu

Guided Practice

tan195

Use half-angle identities to find an exact value of the givenexpression.

1 cos390sin390

1 cos30sin 30

1 3 21 2

2 3

Guided Practice

5sin12

Use half-angle identities to find an exact value of the givenexpression.

1 cos 5 62

cos 5 612 2

1 32 4

1 2 34

1 2 32

Since , we take the positive value…5 1sin 2 312 2

5sin 012

Guided Practice

cos8

Use half-angle identities to find an exact value of the givenexpression.

1 cos 42

1 21

2 2

1 2 24

1 2 22

Since , we take the positive value…1cos 2 2

8 2

cos 08

Guided Practice

sin 2 cos

Write the given expression as one involving only and .

(Note: There are multiple correct answers!!!)

2sin cos cos

cos 2sin 1

sin cos

Guided Practice

sin 2 cos3 sin 2 cos 2 2sin cos cos 2 cos sin 2 sin

2 2 22sin cos cos sin cos 2sin cos 3 22sin cos cos 3sin cos

Write the given expression as one involving only and .

(Note: There are multiple correct answers!!!)

sin cos

Guided Practice2cos6 2cos 3 1x x Prove the identity:

cos6x cos 2 3x22cos 3 1x

Guided Practice2cot 2 cot tanx x x Prove the identity:

2cot 2x2

tan 2x

2

22 tan1 tan

xx

22 1 tan

2 tan

x

x

1 tantan

xx

cot tanx x

Guided Practice 2sin 3 sin 3 4sinx x x Prove the identity:

sin 3x sin 2x x sin 2 cos cos 2 sinx x x x

2 22sin cos 1 2sin sinx x x x

2 2sin 2cos 1 2sinx x x

2 2sin 2 2sin 1 2sinx x x

2sin 3 4sinx x

Guided Practice 2sin 4 4sin cos 2cos 1x x x x Prove the identity:

sin 4x sin 2 2x

2sin 2 cos 2x x

22 2sin cos 2cos 1x x x

24sin cos 2cos 1x x x