Proposition 8.4.13: Taylor Series for the Sine Function
sin(x) = x - 1/3! x3 + 1/5! x5 - 1/7! x7 + ... = for all x
The proof is analogous to the proof for the cos-function. Please look up that proof and adjust it to this situation.
- sin(0) = 0
- sin is an odd function, i.e. sin(-x) = -sin(x)
- sin(x - /2) = -cos(x)
- sin(x) = cos(x)
The fun facts are all easy and are left as exercises. For the third fact you might start with f(x)=cos(x) and develop that in a series around c = /2.