5.5 Multiple-Angle and Product-to-Sum Formulas

Double-Angle Formulas

sin2 2sin cosu u u 2

2 tantan 2

1 tan

uu

u

2 2

2

2

cos2 cos sin

2cos 1

1 2sin

u u u

u

u

Example: Solve 2cos sin2 0x x

Example: If5

cos13

, with theta in the fourth quadrant, find sin 2 ,cos2 , and tan 2 .

Power-Reducing Formulas

2 1 cos2sin

2

uu

2 1 cos2

cos2

uu

2 1 cos2

tan1 cos2

uu

u

Half-Angle Formulas

1 cos

sin2 2

u u

1 coscos2 2

u u

1 cos sintan2 sin 1 cos

u u u

u u

The signs of the sine and cosine depend on the quadrant that u/2 lies in.

Example: Find the exact value ofsin105 .

Example: Find all solutions of 2 22 sin 2cos2

xx in the interval [0,2 ) .

Product-to-Sum Formulas

1

sin sin cos cos2

u v u v u v 1

cos cos cos cos2

u v u v u v

1

sin cos sin sin2

u v u v u v 1

cos sin sin sin2

u v u v u v

Sum-to-Product Formulas

sin sin 2sin cos2 2

u v u vu v

sin sin 2cos sin

2 2

u v u vu v

cos cos 2cos cos2 2

u v u vu v

cos cos 2sin sin

2 2

u v u vu v

Example: Find the exact value ofcos195 cos105 .