Finding fourier coefficients

Question

I need help with finding the Fourier coefficients $a_n$ and $b_n$ of the function $f(t) = \sin(\omega t)$ with a period of $T$. Which coefficients are non-zero?

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I would be very thankful with every help I could get, I am very new to this area.

Answer 1

A periodic function $f(t) $with period $T$ can be written as: $$ f\left(t\right)=\sum_{n=-\infty}^{\infty}\hat{f}\left(n\right)e^{\frac{2\pi j\cdot n}{T}t} $$ where $$\hat{f}(n) =\frac{1}{T}\int\limits _{-\frac{T}{2}}^{\frac{T}{2}}f\left(x\right)e^{-\frac{2\pi nx}{T}}dx $$

The fourier series is unique to each function.

Because of that and Euler's formula we can use the fact that $$ \sin\left(\omega_{0}t\right)=\frac{1}{2j}\left(e^{j\omega_{0}t}-e^{-j\omega_{0}t}\right) $$ Which means: $$ \sin\left(\omega_{0}t\right)=\sin\left(\frac{2\pi}{T}t\right)=\sum_{n=-\infty}^{\infty}\hat{f}\left(n\right)e^{\frac{2\pi}{T}jnt}:\hat{f}\left(n\right)=\begin{cases} \frac{1}{2j} & n=1\\ -\frac{1}{2j} & n=-1\\ 0 & else \end{cases} $$