GATE (TF) Textile 2009 Question Paper Solution | GATE/2009/TF/18

The coefficient of $cos4x$ in the Fourier Series of the function

$f(x)=\begin{cases}-1, & \text{ when } -\pi <x<0 \\ 1, & \text{ when } 0<x< \pi \end{cases} and \;\; f(x)=f(x+2\pi)$

, for all $x$ is

[Show Answer]

Option A is correct

$f(x)=\begin{cases}-1, & \text{ when } -\pi <x<0 \\ 1, & \text{ when } 0<x<\pi \end{cases} and \;\; f(x)=f(x+2\pi)$

By fourier series-

f(x)= $\frac{a_0}{2}+a_1 cosx+a_2 cos2x+a_3 cos3x+a_4 cos4x+..........a_n cosnx+....+\frac{b_0}{2}+b_1 sinx+.....b_n sinnx$

Where,
$a_0=\frac{1}{\pi}\int_{-\pi}^{\pi} f(x) cos nx dx$

$a_0=\frac{2}{\pi}\int_{0}^{\pi} 1 cos nx dx$

$a_0=\frac{2}{\pi}[\frac{1}{n} sin nx]_{0}^{\pi}$

$a_0=2$

Similarily-

$b_n=\frac{1}{\pi}\int_{-\pi}^{\pi} f(x) b_n sin nx dx$
$b_n=0$

and

$a_n=\frac{1}{\pi}\int_{-\pi}^{\pi} f(x) a_n cos nx dx$
$a_n=0$

By putting these values in function, we get-

Coefficient of cos 4x is $a_4$

And all a1,a2,a3,a4….are equal to 0 (Ans)

Option A is correct.