Question

Find the exact value of tan^-1 (-root 3/3)
answer in radians in terms of π

Answer 1

To find
`\tan^(-1)\left(-( √(3) )/(3)\right)` start with range of the function
`y=\tan^(-1)x`. This function has range
`y\in \left[-(\pi)/(2),(\pi)/(2)\right]`.

Since
`-( √(3) )/(3)\ \textless \ 0`, then the angle
`\tan^(-1)\left(-( √(3) )/(3)\right)` lies in VIth quadrant.

It is well known that
`\tan (\pi)/(6)=( √(3) )/(3)`, then
`\tan^(-1)\left(-( √(3) )/(3)\right)=-(\pi)/(6)` (the angle
`-(\pi)/(6)` lies in the VIth quadrant).