Question

How do you evaluate `tan^-1(-1)` without a calculator?

Answer 1

Depending upon if the domain is restricted to `((-pi)/2,pi/2)` the solution will be `pi/4` or if the domain is not restricted the solution will be `pi/4+pik` where `k` is an interger `>=1`

Explanation:

What you are trying to solve is

`x=tan^-1(-1)`

Since the problem deals with the inverse tangent, that means we will use the tangent function to solve the problem.

`tan(x)=tan(tan^-1(1))`

`tan(x)=1`

Now let's think about when `tan(x)=1`

If we are rotating an angle and dealing with a right triangle, we know that `tan(45^@)=1/1=1` and `tan(225^@)=(-1)/-1=1`

Since you are more than likely discussing radians, `x=pi/4`, `x=(5pi)/4`, and we can continue to add `pik` where `k` is an integer`>=1`

Therefore `x=pi/4+pik` where `k` is an interger `>=1`.

If the domain is restricted to `((-pi)/2,pi/2)` then the solution will be `x=pi/4`.