Find the period of \(f(x)=\sin(3x)\).

A function has period \(4\pi\). Find its fundamental angular frequency \(\omega_0\).

Using the convention \(f(x)\sim \frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos nx+b_n\sin nx)\) on \([-\pi,\pi]\), find \(a_0\) for \(f(x)=1\).

Using the same Fourier convention on \([-\pi,\pi]\), find \(a_2\) for \(f(x)=\cos(2x)\).

Using the same Fourier convention on \([-\pi,\pi]\), find \(b_3\) for \(f(x)=\sin(3x)\).

For \(f(x)=x^2\) on \([-\pi,\pi]\), explain which Fourier sine coefficients \(b_n\) are zero.

For \(f(x)=x\) on \([-\pi,\pi]\), explain which Fourier cosine coefficients are zero.

Let \(f(x)=1\) for \(0<x<\pi\) and \(f(x)=-1\) for \(-\pi<x<0\), extended with period \(2\pi\). Find the sine coefficients \(b_n\).

Find the half-range cosine series coefficients for \(f(x)=1\) on \(0<x<L\), using \(f(x)\sim \frac{A_0}{2}+\sum_{n=1}^{\infty}A_n\cos\left(\frac{n\pi x}{L}\right)\).

For the square wave with \(f(x)=1\) on \(0<x<\pi\) and \(f(x)=-1\) on \(-\pi<x<0\), a learner says the Fourier series must equal \(1\) at \(x=0\). Explain the error and give the Fourier series value at \(x=0\).