Question

How do you find the exact value of `tan^-1(tan((2pi)/3))`?

Answer 1

These are inverses of each other. `tan^(-1)(x)` (or `arctanx`) is the inverse of `tan(x)`.

Let `A(x)` be a function, and let `A^(-1)(x)` be its inverse (note that this is not the same as the reciprocal).

Then, the function composition of `A^(-1)(x)` with `A(x)` is `A^(-1)(A(x)) = x`.

Since the domain of `tanx` is `((-pi)/2,pi/2) pm pik` (where `k` is in the set of integers), and the period is `pi`, take the coterminal angle to be `-pi/3`.

So the exact answer is `-pi/3`.