Defining multiples of principal-region angles in inverse trig terms

Question

Consider $tan^{-1}x = \theta$. Then apparently $3\theta=tan^{-1}{}\frac{3x-x^{3}}{1-3x^{2}}$.

How does this work out if $3\theta$ doesn't end up in the first or third quadrant, i.e., if it's not in the range of arctan?

Answer 1

The arctangent is a multivalued function since $\tan$ is periodic with period $\pi$. Thus, to get the correct value of $3\theta$, we have to add or subtract $\pi$ repeatedly.