Question

If $\dfrac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx)$ is the Fourier cosine series of the function $f(x)=\sin x,\; 0<x<\pi$, then which of the following are TRUE?

• A. $a_0 + a_1 = \dfrac{4}{\pi}$
• B. $a_0 = \dfrac{4}{\pi}$
• C. $a_0 + a_1 = \dfrac{2}{\pi}$
• D. $a_1 = \dfrac{2}{\pi}$

Hint:

For cosine series of odd functions like $\sin x$ on $(0,\pi)$, all $a_n$ for $n\ge 1$ vanish except $a_0$.

Answer 1

The Correct Option is A, B

For a Fourier cosine series on $(0,\pi)$, the coefficients are:
\[ a_0 = \frac{2}{\pi} \int_0^\pi f(x)\,dx, \qquad a_n = \frac{2}{\pi} \int_0^\pi f(x)\cos(nx)\,dx. \]
Given $f(x)=\sin x$, compute $a_0$:
\[ a_0 = \frac{2}{\pi}\int_0^\pi \sin x\,dx = \frac{2}{\pi}\left[-\cos x\right]_0^\pi = \frac{2}{\pi}(2)=\frac{4}{\pi}. \]
Thus, statement (B) is correct.
Next, compute $a_1$:
\[ a_1=\frac{2}{\pi}\int_0^\pi \sin x \cos x\,dx =\frac{1}{\pi}\int_0^\pi \sin(2x)\,dx. \]
\[ \int_0^\pi \sin(2x)\,dx = 0, \]
so $a_1=0$. Thus, (D) is false.
Since $a_1=0$,
\[ a_0 + a_1 = \frac{4}{\pi}. \]
Hence, statement (A) is correct. (C) is false.