Chapter 4, Problem 82RE

Solution

To Find: The exact value of $x$ without using a calculator.

The solution is $x=\frac{5\pi }{6}$ .

Given information:

The $\cot x=-\sqrt{3},0\le x\le \pi$

Calculation:

Solve for $x$

  $\begin{array}{l}\cot x=-\sqrt{3},0\le x\le \pi \text{ Given}\\ \cot x=-\sqrt{3}\\ \frac{1}{\tan x}=-\sqrt{3}\\ \tan x=-\frac{1}{\sqrt{3}}\end{array}$

  $\begin{array}{l}\tan x=-\tan \frac{\pi }{6}\left[\because 0\le x\le \pi \right]\\ \tan x=\tan \left(\pi -\frac{\pi }{6}\right)\left(\because -\tan x=\tan \left(\pi -x\right)\right)\\ \tan x=\tan \left(\frac{5\pi }{6}\right)\\ x=\frac{5\pi }{6}\end{array}$

Hence, $x=\frac{5\pi }{6}$ .