Cosine Formulas

cos(a + b) = cos a cos b - sin a sin b

cos(a - b) = cos a cos b + sin a sin b

Sine Formulas

sin(a + b) = sin a cos b + cos a sin b

sin(a - b) = sin a cos b - cos a sin b

sin2a = 2 sin a cos a

Tangent Formulas

tan(a +b) = (tan a + tan b)/(1 - tan a tan b)

tan(a - b) = (tan a -tan b)/(1 + tan a tan b)

Derivations

• cos(a + b) = cos a cos b - sin a sin b
• cos(a - b) = cos a cos b + sin a sin b

Using cos(a + b) and the fact that cosine is even and sine is odd, we have

            cos(a + (-b)) = cos a cos (-b) - sin a sin (-b)
                          = cos a cos b - sin a (-sin b)
                          = cos a cos b + sin a sin b

• sin(a + b) = sin a cos b + cos a sin b

Using cofunctions we know that sin a = cos (90 - a). Use the formula for cos(a - b) and cofunctions we can write

         sin(a + b) = cos(90 - (a + b)) 
                    = cos((90 - a) - b)
                    = cos(90 -a)cos b + sin(90 - a)sin b
                    = sin a cos b + cos a sin b

• sin(a - b) = sin a cos b - cos a sin b

Having derived sin(a + b) we replace b with "-b" and use the fact that cosine is even and sine is odd.

      sin(a + (-b)) = sin a cos (-b) + cos a sin (-b) 
                    = sin a cos b + cos a (-sin b)
                    = sin a cos b - cos a sin b