10-1

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

There are two main formulas you will use in the section. They are the 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

There are four other formulas for this section that aren't used as often as the main two. These formulas are used when 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)

Here are some examples:
Example 1: Find the exact value of 75.

• cos (45 + 30)


• cos 45 cos 30 - sin 45 sin 30 *Now, you change from degrees to radians by using the trig chart.


• (sq. root of 2/2) (sq. root of 3/2) - (sq. root of 2/2) (1/2)


• (sq. root of 6 - sq. root of 2)/4

Example 2: Simplify the given expression, cos 105 cos 15 + sin 105 sin 15.

• cos (105 - 15)


• cos 90


• 0

Example 3: Prove that the equation, cos (pi + x) = -sin x.

• cos pi cos x - sin pi sin x


• -cos x - 0


• -cos x