How do I find the Fourier Series for F(t) = sin(wt)?

[physicspupil]

Homework Statement

Consider F(t) = sin(wt) when 0 < t < pi/w and 0 when pi/w < t < 2pi/w. Where w is the frequency and t is the time. Find the Fourier Series

Homework Equations

F(t) = sum of (ck e^ikt)

See attached doc with math type; its a lot more readable.

The Attempt at a Solution

ck = (w/2pi integral from 0 to pi/w of sin(wt)*e^-ikx) + (w/2pi integral from pi/w to 2pi/w of sin(wt)*e^-ikx)

I’m not sure how to do this integration. If all the trig functions had the same argument, it would be do-able, but they don’t. w is a constant, right? And k isn’t, right? We never dealt with stuff like this back in calc. I’m very grateful for any help or suggestions. I tried to use trig addition formulas, but that didn’t work. Thanks!

 

[Meir Achuz]

Your integral is over t, not x, so it should have exp(-ikt).
Write sin(wt) as [exp(iwt)-exp(-iwt)]/2i, and do the exponential integrals.