If we know $\sin \alpha$ and $\cos \alpha.$ can we find the angle $\alpha$?

Question

For example $$\sin \alpha=\frac{8}{\sqrt{65}}, \cos \alpha=\frac{1}{\sqrt{65}}$$

Can we analytically find $\alpha$ ? The only thing I did was calculate $\tan$; it isn't helpful.

It turns out my question is if we know $\cos \alpha $ and $\sin \alpha$ is there a formula for $\arcsin$ and $\arccos$ ? The only way of finding arcsin or arccos that i know is using a calculator

Answer 1

There are infinitely many such $\alpha$'s. However, you can get one such $\alpha$ taking$$\alpha=\arcsin\left(\frac8{\sqrt{65}}\right)=\int_0^{\frac8{\sqrt{65}}}\frac{\operatorname dx}{\sqrt{1-x^2}}.$$

Answer 2

My calculator gives $1.4464...$

From Taylor series $\alpha=\arccos(\dfrac 1 {\sqrt{65}})=\dfrac\pi2-\dfrac 1{\sqrt{65}}...\approx\dfrac\pi2-\dfrac18\approx1.4458.$

Answer 3

There is a Taylor series expansion for $\arcsin(x)$ as described here:

Maclaurin expansion of $\arcsin x$

Then $\arccos(x) = {\pi \over 2} - \arcsin(x)$ in the first quadrant.