Say we know for a fact that $$ \int_0^{2\pi} u(r,\theta)r d\theta = 0 $$
can we show that $$ \int_0^{2\pi}u^2\sin(\theta)d\theta = 0. $$
I tried using a past question yesterday on stack exchange using properties of functions that are $0$ on an interval, since I know $\sin$ is $0$ integrated over a full period. Upon trying integration by parts it also led nowhere.
I don't want a full solution just a hint would help!
Thank you!
Hint: Check $u(r,\theta) = \theta^2 - \frac{4\pi^2}{3}$.