trigonometry - Alternative of finding theta when sin $theta$ and cos $theta$ are given

For example, we're given a problem in which sin $\theta = \sqrt3/2$ and cos $\theta = -1/2$. To find out the angle $\theta$, I look at the unit circle and I get the answer. However, I was just curious whether there's an alternative to this, any idea? Because when I tried using cos(-$\theta$) = cos$\theta$, I get the wrong value of $\theta$ as we've been provided with the value of sin $\theta$ as well...

Answer

$$\sin\theta=\frac{\sqrt3}2=\sin\frac\pi3\implies\theta=n\pi+(-1)^n\frac\pi3$$
where $n$ is any integer

Set $n=2s+1,$(odd) $=2s$(even) one by one

Again,
$$\cos\theta=-\frac12=-\cos\frac\pi3=\cos\left(\pi-\frac\pi3\right)$$

$$\implies\theta=2m\pi\pm\left(\pi-\frac\pi3\right)$$
where $m$ is any integer

Check for '+','-' one by one

Observe that the intersection of the above two solutions is $$\theta=2r\pi+\frac{2\pi}3$$
where $r$ is any integer