The Unit Circle and Trignomoetric Functions

Question

Use the Sum Formula for Cosine to derive the following identity for the average rate of change of the cosine function:

cos (x + h) − cos x /h = cos x (cos h − 1)/h − sin x (sin h)/h

Note: You will learn in Calculus 1 that limh→0(cos h−1)/h = 0 and that limh→0 sin h/h = 1. It follows that the instantaneous rate of change of cos x is − sin x.

Answer 1

Sum Formula for Cosine -> cos(A+B) = cos(A)cos(B) - sin(A)sin(B)

1) [cos(x + h) - cos(x)] / h *AROC of cosine function

2) [cos(x)cos(h) - sin(x)sin(h) - cos(x)] / h *Used sum formula for cosine to expand cos(x + h)

3) [cos(x)cos(h) - cos(x) - sin(x)sin(h)] / h *Grouped like terms

4) [cos(x)[cos(h) - 1] - sin(x)sin(h)] / h *Distributive property, pull out cos(x) factor from like terms

5) cos(x)[(cos(h) - 1) / h] - sin(x)[sin(h) / h] *Finished :)

Hope this helps!