## Saturday, February 3, 2007

### Precalculus 5.4 Sum and Difference Formulas

5.4 Sum and Difference Formulas

Sum and Difference Formulas

sin (u + v) = sin u cos v + cos u sin v
sin (u - v) = sin u cos v - cos u sin v

cos (u + v) = cos u cos v - sin u sin v
cos (u - v) = cos u cos v + sin u sin v

A) Evaluating a Trigonometric Function:
1. Example 1

sin 75° = sin (30° + 45°)
sin 75° = sin 30° cos 45° + cos 30° sin 45°

Example 2: sin 90° = 1

sin 90° = sin (30° + 60°)
sin 90° = sin 30° cos 60° + cos 30° sin 60°

B) Proving a Cofunction Identity

Example 3:

cos (π - θ ) + sin (π/2 + θ ) = 0

cos π cos θ + sin π sin θ + sin (π/2) cos θ + cos (π/2) sin θ = 0

(-1) cos θ + (0)(sin θ ) + (1) cos θ + (0) sin θ = 0

0 = 0

Example 4:
(Cos ( x + h) - cos x)/h = (cos x (cos (h) - 1))/h - (sin x sin h)/h

(Cos x cos h - sin x sin h - cos x)/ h = (cos x (cos (h) - 1))/h - (sin x sin h)/h

(Cos x ( cos (h) - 1) - sin x sin h)/h = (cos x (cos (h) - 1))/h - (sin x sin h)/h

(cos x (cos (h) - 1))/h - (sin x sin h)/h = (cos x (cos (h) - 1))/h - (sin x sin h)/h

C) Solving a Trigonometric Equation:

Example 5:

cos (x + π/6) - cos (x - π/6) = 1

cos x cos π/6 - sin x sin π/6 -(cos x cos π/6 + sin x sin π/6) = 1

cos x cos π/6 - sin x sin π/6 - cos x cos π/6 - sin x sin π/6 = 1

-2 sin x (½) = 1
-1 sin x = 1
sin x = -1

x = 3π/2

5.4 Homework #46: pg 408; # 3, 5, 11, 15, 19 - 27 odd, 35 - 57 odd