To give some background to the question, I am preparing for an admissions test for a physics course at university (the PAT), and for this test we are not allowed to use calculators or tables in any form.
In the maths section of the test, there are often trigonmetric equations such as the one below which we have to solve for $x$ within a certain range (usually $[0,\pi]$):
Solve $\sin{3x}=\sqrt{3}\cos{3x}$, for $x$ in the range $0\leq x \leq \pi$.
This problem is easy to reduce:
$\tan{3x}=\sqrt{3}\implies3x=\tan^{-1}{\sqrt{3}}\implies x=\frac{\tan^{-1}{\sqrt{3}}}{3}$
However, I am unsure how I can evaluate expressions such as $\tan^{-1}{\sqrt{3}}$ without a calculator, is there any method I can use for expressions like these, or am I required to simply learn various common values of $\tan$ (i.e. $\tan{\pi}$, $\tan{\frac{\pi}{4}}$, $\tan{0}$, etc.)?
Thanks in advance!