Uniqueness of periodic functions

Question

Given a periodic function $f(x)$, how many criteria would it have to meet to be able to with certainty identify it with some known periodic function, i.e. $ \sin(x) $? What prompted this question for me was a problem for my calculus class, where we've been given a function $f(x)$ with the following properties: $$f(x) = f(x+2\pi)$$ $$ \int_0^\pi f(x)\,dx = 2$$ $$f(-x) = -f(x)$$ $$f(x) \geq 0 \text{ for } 0 \leq x \leq \pi $$ $$\text{and } f\left(\frac{\pi}{2}\right) = 1$$ Is this enough to conclude that $f(x)=\sin(x)$? If so, what theorem proves this? And if not, what criteria would the function have to meet?