Chapter 1.6, Problem 7QR

To determine

To find: The solution of the equation $\tan x=1$ graphically in the given interval $-\frac{\pi }{2}\le x\le \frac{3\pi }{2}$ .

Answer to Problem 7QR

The solution of the equation $\tan x=1$ in the interval $-\frac{\pi }{2}\le x\le \frac{3\pi }{2}$ is $x\approx 0.785$ and $x\approx 3.93$ .

Explanation of Solution

Given information:

The given equation is $\tan x=1$ with the interval $-\frac{\pi }{2}\le x\le \frac{3\pi }{2}$ .

Calculation:

The solutions of given equation $\tan x=1$ are the intersecting points of both the curves $y=\tan x$ and $y=1$ for $-\frac{\pi }{2}\le x\le \frac{3\pi }{2}$ .

To graph function $y=\tan x$ and $y=1$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\tan x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{TAN}}\boxed{\text{X}}\boxed{)}$ . Now press the $\boxed{\text{ENTER}}$ key and enter right hand side of the equation $y=1$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{1}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\tan \left(\text{X}\right)\\ {\text{Y}}_2=1\end{array}$

Press the $\boxed{\text{WINDOW}}$ key and adjust the value of $x$ to $\left[-\frac{\pi }{2},\frac{3\pi }{2}\right]$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of both the equations as shown. To find the intersection points of both the curves enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{TRACE}}\boxed{5}\boxed{\text{ENTER}}\boxed{\text{ENTER}}\boxed{\text{ENTER}}$ .

  

Figure (1)

As observed from the graph both the curves intersect at two points $x\approx 0.785$ and $x\approx 3.93$ .

Therefore, the solution of the equation $\tan x=1$ in the interval $-\frac{\pi }{2}\le x\le \frac{3\pi }{2}$ is $x\approx 0.785$ and $x\approx 3.93$ .