Orthonormality of $\sin(nx)/\sin x$ under a weighted inner product

Let $C[0,\pi]$ be the real vector space of real-valued continuous functions on the closed interval $[0,\pi]$. For positive integers $n$, define $f_n\in C[0,\pi]$ by

$$f_n(x)=\begin{cases}\dfrac{\sin(nx)}{\sin x}&\text{if }x\in(0,\pi),\ n&\text{if }x=0,\ (-1)^{n-1}n&\text{if }x=\pi.\end{cases}$$

Let $V$ be the real subspace of $C[0,\pi]$ spanned by $\{f_1,f_2,f_3\}$. Consider the inner product on $V$ given by

$$\langle f,g\rangle=\frac2\pi\int_0^\pi f(x)g(x)\sin^2x\,dx.$$

Which of the following statements is true?