# Thread: Finding the sum of a trigonometric series?

1. ## Finding the sum of a trigonometric series?

Find the sum of the series . You must calculate the sum of this series only by multiplying through by

Now I've heard of finding the sum of a trig series by finding real and imaginary parts etc., but I have no idea how to do it this way.

any help or advice is hugely appreciated!

thank you

2. ## Re: Finding the sum of a trigonometric series?

Make repeated use of the trig identity

$\sin(A)-\sin(B)=2\cos\left(\frac{A+B}{2}\right)\sin \left( \frac{A-B}{2}\right).$

3. ## Re: Finding the sum of a trigonometric series?

Originally Posted by BobP
Make repeated use of the trig identity

$\sin(A)-\sin(B)=2\cos\left(\frac{A+B}{2}\right)\sin \left( \frac{A-B}{2}\right).$
Hi Bob, thank you for that

could you possibly explain this further? would I do this by substituting?

4. ## Re: Finding the sum of a trigonometric series?

Having multiplied throughout by $2\sin(x/2),$ consider a general term in the resulting series, $2\cos(kx)\sin(x/2).$

Use the trig identity to break this into two terms.

Apply this to each term of your series in turn, when you've done this, you should find that there is a whole lot of cancellation, after which you need to use the trig identity again.
Don't forget to divide the resulting expression by $2\sin(x/2),$ .