If $f:[0, \pi] \rightarrow \mathbb{R}$ is continuous, $f(0) = 0$, then does $\int_0^\pi f(t)\cos nt\,dt = 0$ for $n=0, 1,2,\dots$ imply $f = 0$?

Question

Let $f:[0, \pi] \rightarrow \mathbb{R}$ is continuous and $f(0) = 0$, then which of the following are true?

1. If $$\int_0^\pi f(t)\cos nt\,dt = 0$$ for $n \in \{0\} \cup \mathbb{N}$, then $f \equiv 0$
2. If $$\int_0^\pi f(t)\sin nt\,dt = 0$$ for $n \in \mathbb{N}$, then $f \equiv 0$
3. If $$\int_0^\pi t^nf(t)\,dt = 0$$ for $\{0\} \cup \mathbb{N}$, then $f \equiv 0$

Answer 1

Hint for (1) and (2): If $f$ is continuous on $[0,\pi]$ then $f$ can be extended to an even continuous function on $[-\pi,\pi]$. If $f$ is continuous on $[0,\pi]$ and $f(0)=0$ then $f$ can also be extended to an odd function continuous on $[-\pi,\pi]$. In either case you can use the given information to find $\int_{-\pi}^\pi f(t)\cos(nt)$ and $\int_{-\pi}^\pi f(t)\sin(nt)$.