Question

Explain why tan pi=0 does not imply that arctan0=pi?

Answer 1

`pi` is not in the interval used for `arctan`

Explanation:

We want `arctan` to be a function. We want it to give one, never two, values for a single input.

By definition:

`y = arctan x` if and only if (`tan y = x` and `-pi/2 < y < pi/2`)

Since `pi` is not in `(-pi/2, pi/2)` there is no `x` for which `arctanx = pi`

(The situation is similar to: Explain why `(-3)^2 = 9` does not imply that `sqrt9 = -3`. It (`-3`) is the wrong kind of number to be a principal square root.)