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A periodic function $f(t)$, with a period of $2 \pi$, is represented as its Fourier series, $$f(t) = a_0 + \sum_{n=1}^{\infty }a_n \cos nt + \sum_{n=1}^{\infty} b_n \sin nt.$$ if $$f(t) = \begin{cases} A \: \sin t, & 0 \leq t \leq \pi \\ 0, & \pi < t < 2 \pi \end{cases}$$

the Fourier series coefficients $a_1$ and $b_1$ of $f(t)$are

  1. $a_1 = \frac{A}{\pi}; \: b_1 = 0$
  2. $a_1 = \frac{A}{2}; \: b_1 = 0$
  3. $a_1 = 0; \: b_1 = \frac{A}{\pi}$
  4. $a_1 = 0;b_1 = \frac{A}{2}$
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