Question

The coefficient $a_4$ of the trigonometric Fouries series of a periodic signal with period of $2\pi$ whose one period is given as $x\left(t\right)=\{\begin{array}{cc}sint,& 0\le t\le \pi \\ 0,& \pi \le t\le 2\pi \end{array}$
Where $a_n$ is coefficient of $cos\left(n{\omega }_{0}t\right)$ in the trigonometric Fourier series expansion.

• A. $-\frac{2}{15\pi }$
• B. $\frac{2}{15\pi }$
• C. 0
• D. $\frac{2}{3\pi }$

Answer 1

The correct option is A $-\frac{2}{15\pi }$

${\omega }_0=\frac{2\pi }{T}=\frac{2\pi }{2\pi }=1$

$a_4=\frac{2}{2\pi }{\int }_0^{\pi }sintcos4t\cdot dt$

$=\frac{1}{2\pi }{\int }_0^{\pi }\left[sin\right(5t)-sin(3t\left)\right]dt$

$=\frac{1}{2\pi }\left[{(\frac{cos5t}{5}-\frac{cos3t}{3})}_{\pi }^0\right]$

$=\frac{2}{2\pi }[\frac{1}{5}-\frac{1}{3}]$

$=\frac{1}{\pi }(-\frac{2}{15})=-\frac{2}{15\pi }$