Examples of orthogonal function bases

Question

These days we've been solving the heat equation in class for the $1$D case of a bar of length $\ell$ with two thermal reservoirs at its ends which have the same temperature, $0^\circ$C. Yesterday we found that the solution (using separation of variables) was $$u(x,t)=\sum_{r=1}^\infty B_r\exp\left(-{\frac{r^2\pi^2t}{\ell^2}}\right)\sin\left(\frac{r\pi x}{\ell}\right).$$ Today, we were left with finding the infinite coefficients using the initial condition $X(x)T(0)=f(x)$, namely, $$u(x,0)=\sum_{r=1}^\infty B_r\sin\left(\frac{r\pi x}{\ell}\right)=f(x).$$ This way, our professor briefly introduced the function inner product, and without further extending, we found that (assuming summation and integral interchangeable) $$\begin{aligned} \left\langle f(x)\;\middle|\;\sin\left(\frac{n\pi x}{\ell}\right)\right\rangle&=\int_0^\ell f(x)\sin\left(\frac{n\pi x}{\ell}\right)\,\mathrm dx\\ &=\sum_{r\geq 1}B_r\int_0^\ell \sin\left(\frac{r\pi x}{\ell}\right)\sin\left(\frac{n\pi x}{\ell}\right)\,\mathrm dx\\ &=\sum_{r\geq 1}B_r\underbrace{\left\langle \sin\left(\frac{r\pi x}{\ell}\right)\;\middle|\;\sin\left(\frac{n\pi x}{\ell}\right)\right\rangle}_{\propto \delta_{rn}}\propto B_n \end{aligned}$$ Meaning that $\left\{\sin\left(\dfrac{r\pi x}{\ell}\right)\right\}_{r=1}^\infty$ is an orthogonal basis. If I don't recall wrong, I saw somewhere that Legendre polynomials were an orthogonal basis too. The goal of this post was basically asking for more examples of orthogonal bases, in other words, functions that obey $$\int f_i(x)f_j(x)\,\mathrm dx\propto\delta_{ij}$$ I'm guessing there are some trigonometric basis similar to the sine's too.