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Find the complex exponential form (i.e. $\sum_{n=-\infty}^{\infty}c_n e^{\frac{2\pi}{T}nt}$) of the Fourier series of $$2+\frac{1}{2}\cos(t+45^\circ)+2\cos(3t)-2\sin(4t+30^\circ)$$

EDIT: Some info on what I've done so far.

My first instinct is to get all of the coefficients. I tried to do this with this integral: $$ c_n=\frac{1}{2\pi}\int_{0}^{2\pi}(2+\frac{1}{2}\cos(t+45^\circ)+2\cos(3t)-2\sin(4t+30^\circ))e^{-j n t}\, \mathrm{d}t $$ But the integral turned out pretty hairy and I'm wondering if I'm going about this wrong.

Dominic Michaelis
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user1539179
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1 Answers1

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First, convert your degrees into radians. Recall that $t$ degrees are worth $$ x=\frac{\pi t}{180} $$ radians.

Then use $$\cos x=\frac{e^{ix}+e^{-ix}}{2}$$ and $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}.$$

No need to integrate, it's been done before to get your Fourier series.

Julien
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