Find when two sin graphs hit $x=0$ at the same time

Question

Lets say I have 2 sine functions. They could be any sine functions but I'm more interested in the ones that get bigger/smaller as they go. So we have: $$ y_1 = \sin{(x^2)} $$ $$ y_2=\sin{\left(\sqrt{x}\right)} $$

I'm interested in when they both hit $y=0$ at the same time.

I don't know where to even start.

Answer 1

You have $x^2 = \pi n$ and $\sqrt{x} = \pi m$. Eliminating $x$ gives $n = \pi^3 m^4$. As $\pi$ is irrational $m$ must be 0, and hence $x = 0$ is the only solution.