Question

In Problems 13–24, find the exact value of each expression.

tan^-1(- 1)

Short answer

The exact value of tan^-1 0 is $-\frac{\mathrm{\pi }}{4}$.

Step-by-step solution

Step 1. Given Information:   

Given expression:tan^-1 (- 1)

We want to find the exact value of given expression.

Step 2. Solution:  

The given expression can be rewritten as:

${\tan }^{-1}(-1)=y;-\frac{\mathrm{\pi }}{2}<y<\frac{\mathrm{\pi }}{2}$

Hence we get tan (y) = - 1

Since we have tan^-1 function initially, the bounds on the tan function are going to be in the interval $\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$ which represents the range of tan^-1 function.

Using the unit circle, we know that in order to get (- 1) , y must be equal to $-\frac{\mathrm{\pi }}{4}$.

Hence, we get ${\tan }^{-1}(-1)=-\frac{\mathrm{\pi }}{4}$.