Chapter 7, Problem 8RE

To determine

The exact value of the expression ${\cot }^{-1}\left(-1\right)$.

Answer to Problem 8RE

Solution:

The exact value of the expression ${\cot }^{-1}\left(-1\right)$ is $\frac{3\pi }{4}$.

Explanation of Solution

Given information:

${\cot }^{-1}\left(-1\right)$.

Explanation:

Let $\theta ={\cot }^{-1}\left(-1\right)$.

Then, $\cot \theta =-1$ and $0<\theta <\pi$.

Since, the domain of function $y={\cot }^{-1}x$ is $\left(-\infty ,\infty \right)$ and its range is the interval $\left(0,\pi \right)$.

Now, $\cot \theta =\frac{1}{\tan \theta }=-1$, this means $\tan \theta =-1$.

Thus, ${\tan }^{-1}\left(-1\right)=\theta$.

$\Rightarrow \tan \theta =1$; $-\frac{\pi }{2}<\theta <\frac{\pi }{2}$.

Using table and figure,

$\begin{array}{cccccccccc}\theta & -\frac{\pi }{2}& -\frac{\pi }{3}& -\frac{\pi }{4}& -\frac{\pi }{6}& 0& \frac{\pi }{6}& \frac{\pi }{4}& \frac{\pi }{3}& \frac{\pi }{2}\\ \tan \theta & \infty & -\sqrt{3}& -1& -\frac{\sqrt{3}}{3}& 0& \frac{\sqrt{3}}{3}& 1& \sqrt{3}& \infty \end{array}$

The only angle $\theta$ in the interval $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ whose $\tan \theta =-1$ is $-\frac{\pi }{4}$.

$\Rightarrow {\tan }^{-1}\left(-1\right)=-\frac{\pi }{4}$.

Hence, for the interval $0<\theta <\pi$, ${\cot }^{-1}\left(-1\right)=\pi -\frac{\pi }{4}=\frac{3\pi }{4}$.

Therefore, the exact value of the expression ${\cot }^{-1}\left(-1\right)$ is $\frac{3\pi }{4}$.