10-1

10-1 Formulas for cos (a +/- B) and sin (a +/- B)

Sum and Difference Formulas for Cosine and Sine:
-cos (a +/- B) = cos a cos B -/+ sin a sin B
-sin (a +/- B) = sin a cos B +/- cos a sin B

Rewriting a Sum or Difference as a Product:
-sin x + sin y = 2 sin (x + y/2) cos (x - y/2)
-sin x - sin y = 2 cos (x + y/2) sin (x - y/2)
-cos x + cos y = 2 cos (x + y/2) cos (x - y/2)
-cos x - cos y = -2 sin (x + y/2) sin (x - y/2)

Example 1: Simplify the given expression.
a) cos 105 cos 15 + sin 105 sin 15
=cos (105 - 15)
=cos (90)
=0
b) sin 3x cos 2x - cos 3x sin 2x
=sin (3x - 2x)
=sin x

Example 2: Find the exact value of each expression.
a) cos 105
=cos (60 + 45)
=cos 60 cos 45 - sin 60 sin 45
=(1/2) (sq root of 2/2) - (sq root of 3/2) (sq root of 2/2)
=sq root of 2/4 - sq root of 6/4
=(sq root of 2 - sq root of 6)/2

-Jordan Duhon