Chapter 5.3, Problem 82E

To determine

To solve: The given equation using inverse trigonometric functions.

Answer to Problem 82E

The solutions of the equation are:

  $\begin{array}{l}x=\arctan 2+n\pi \\ x=-\frac{\pi }{4}+n\pi \end{array}$

Explanation of Solution

Given information:

The following equation:

  ${\tan }^{2}x-\tan x-2=0$

Calculation:

The equation is solved as follows:

  $\begin{array}{l}{\tan }^{2}x-\tan x-2=0\\ (\tan x-2)(\tan x+1)=0\text{ [Factor]}\end{array}$

Setting each factor equal to zero, two solutions are obtained in the interval $(-\pi /2,\pi /2)$

[Range of inverse tangent function is $(-\pi /2,\pi /2)$ ]

  $\begin{array}{l}\tan x-2=0\\ \tan x=2\text{ [Adding 2 to both sides]}\\ x=\text{arctan 2 [Inverse tangent function]}\end{array}$

and

  $\begin{array}{l}\tan x+1=0\\ \tan x=-1\text{ [Subtracting 1 from both sides]}\\ x=\text{arctan(-1) [Inverse tangent function] }\\ x=-\frac{\pi }{4}\text{ [Value of arctan(-1)}]\end{array}$

Finally because $\tan x$ has a period of $\pi$ , add multiples of $\pi$ to obtain,

  $x=\arctan 2+n\pi$ and $x=-\frac{\pi }{4}+n\pi$ [General solution]

Where $n$ is an integer.

The solution is confirmed graphically as follows:

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression ${\tan }^{2}x-\tan x-2$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as: