Taylor series os sinxcosx

Giruilla

Wondering would anybody be able to help with this problem.

I know the Taylor series of sinx = x - x^3/3! + x^5/5! etc
cosx = x - x^2/2! etc

I there any easy representation sinxcosx like this or do you simply have to multiply them out? Also similarly is there any similar representation of (sinx)^2?

seandoiler

well the taylor series for sin[x]cos[x] is
x- 4/3! x^3 + 16/5! x^5 - 64/7! x^7 + ...
there are two ways to get it...(1) multiply taylor for sin[x] by taylor for cos[x] or alternatively (2) use the taylor expansion and you see you need to calculate the value of the nth derivative of sin[x] cos[x] evaluated at 0....a pattern will soon emerge and hey presto....

by the way the taylor for cos[x] is 1 - 1/2! x^2 + ... not x as you have written above

for sin[x] ^2 same as for sin[x] cos[x]...either mulitply or calculate derivative evaluated at zero and look for pattern

hope this helps

sean

Giruilla

That helps a lot thanks sean:)