# COMPOUND ANGLE FORMULAS FOR SINE AND COSINE

## Compound Angle Formulas

sin (A + B) = sin A cos B + cos A sin B

sin (A - B) = sin A cos B - cos A sin B

cos (A + B) = cos A cos B - sin A sin B

cos (A - B) = cos A cos B + sin A sin B

Problem 1 :

Express each of the following in the form sin α, where α is acute.

a) sin 10° cos 30° + cos 10° sin 30°

Solution:

sin 10° cos 30° + cos 10° sin 30°

sin (A + B) = sin A cos B + cos A sin B

= sin (10° + 30°)

= sin 40°

b) sin 67° cos 18° - cos 67° sin 18°

Solution:

sin 67° cos 18° - cos 67° sin 18°

sin (A - B) = sin A cos B - cos A sin B

= sin (67° - 18°)

= sin 49°

c) sin 62° cos 74° + cos 62° sin 74°

Solution:

sin 62° cos 74° + cos 62° sin 74°

sin (A + B) = sin A cos B + cos A sin B

= sin (62° + 74°)

= sin 136°

d) cos 14° cos 39° - sin 14° sin 39°

Solution:

cos 14° cos 39° - sin 14° sin 39°

cos (A + B) = cos A cos B - sin A sin B

= cos (14° + 39°)

= cos 53°

Problem 2 :

Express as a single trigonometric ratio.

a) cos A cos 2A - sin A sin 2A

Solution:

cos A cos 2A - sin A sin 2A

cos (A + B) = cos A cos B - sin A sin B

= cos (A + 2A)

= cos 3A

b) sin 4A cos B - cos 4A sin B

Solution:

sin 4A cos B - cos 4A sin B

sin (A - B) = sin A cos B - cos A sin B

= sin (4A - B)

c) cos A cos 3A + sin A sin 3A

Solution:

cos A cos 3A + sin A sin 3A

cos (A - B) = cos A cos B + sin A sin B

= cos (A - 3A)

= cos (-2A)

= cos 2A

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