Question about the argument of a complex number

Question

I was trying to do some roots of a complex number and I have $$\text{argument}= \arctan\left(\frac{\sqrt3+1}{\sqrt3-1}\right)$$ How do you actually find the value in radians. From solutions I find $5\pi/12$, but in an exam, how are you supposed to find that? By approximations, like thinking how much is arctan of $1.73$?

Answer 1

Let $\theta= \arctan\left(\frac{\sqrt3+1}{\sqrt3-1}\right)$. Then,

$$\tan\theta= \frac{\sqrt3+1}{\sqrt3-1}=\frac{1+\frac1{\sqrt3}}{1-1\cdot\frac1{\sqrt3}} =\frac{\tan45+\tan30}{1-\tan45\cdot\tan30}=\tan(45+30)=\tan75$$