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

Dec 1, 2016

Depending upon if the domain is restricted to $\left(\frac{- \pi}{2} , \frac{\pi}{2}\right)$ the solution will be $\frac{\pi}{4}$ or if the domain is not restricted the solution will be $\frac{\pi}{4} + \pi k$ where $k$ is an interger $\ge 1$

#### Explanation:

What you are trying to solve is

$x = {\tan}^{-} 1 \left(- 1\right)$

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

$\tan \left(x\right) = \tan \left({\tan}^{-} 1 \left(1\right)\right)$

$\tan \left(x\right) = 1$

Now let's think about when $\tan \left(x\right) = 1$

If we are rotating an angle and dealing with a right triangle, we know that $\tan \left({45}^{\circ}\right) = \frac{1}{1} = 1$ and $\tan \left({225}^{\circ}\right) = \frac{- 1}{-} 1 = 1$

Since you are more than likely discussing radians, $x = \frac{\pi}{4}$, $x = \frac{5 \pi}{4}$, and we can continue to add $\pi k$ where $k$ is an integer$\ge 1$

Therefore $x = \frac{\pi}{4} + \pi k$ where $k$ is an interger $\ge 1$.

If the domain is restricted to $\left(\frac{- \pi}{2} , \frac{\pi}{2}\right)$ then the solution will be $x = \frac{\pi}{4}$.