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
