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Fourier series and integration watch

1. I've got partway through this question and i feel like im so close to finishing it but just having a few problems along the way:

The question is:

Express the 2pi periodic function F(t)=Cos(t)sin(4t) as a finite fourier series...

So far i have managed to change the f(t) into a function of sin
f(t)=1/2 (sin(5t)+sin(3t))

from here on i believe the next step is:
Bn= (1/pi) (Integral between 0 and pi of sin(5t)sin(nt)+sin(3t)sin(nt))dt

im unsure as to whether the next step is to use the product of 2 sines trig identity, but then i end up with a big integral in terms of cos with constants n and t and it all gets very confusing from here.

could anyone provide any help?
2. (Original post by SassyPete)
I've got partway through this question and i feel like im so close to finishing it but just having a few problems along the way:

The question is:

Express the 2pi periodic function F(t)=Cos(t)sin(4t) as a finite fourier series...

So far i have managed to change the f(t) into a function of sin
f(t)=1/2 (sin(5t)+sin(3t))
This *is* a finite Fourier series. You don't need to do anything else.
3. (Original post by DFranklin)
This *is* a finite Fourier series. You don't need to do anything else.
do i not need to integrate?
4. (Original post by SassyPete)
do i not need to integrate?
No. (It is *possible* your lecturer may want you to integrate it as an exercise, but mathematically it's not needed).

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