Chapter 1.6, Problem 34E

To determine

To find: The solution of the equation $\sec x=-3$ graphically in the given interval $-\pi \le x<\pi$ .

Answer to Problem 34E

The solution of the equation $\sec x=-3$ in given interval are $x\approx -1.911$ and $x\approx 1.911$ .

Explanation of Solution

Given information:The given equation is $\sec x=-3$ with the interval $-\pi \le x<\pi$ .

Calculation:

The solutions of given equation $\sec x=-3$ are the intersecting points of both the curves $y=\sec x$ and $y=-3$ for $-\pi \le x<\pi$ .

To graph a function $y=\sec x$ and $y=-3$ , 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 $y=\sec x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{1}\boxed{÷}\boxed{\text{COS}}\boxed{\text{X}}\boxed{)}$ . Now press the $\boxed{\text{ENTER}}$ key and enter right hand side of the equation $y=-3$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{(-)}\boxed{3}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=1/\cos \left(\text{X}\right)\\ {\text{Y}}_2=-3\end{array}$

Press the $\boxed{\text{WINDOW}}$ key and adjust the window to $\left[-\pi ,\pi \right]$ by $\left[-5,5\right]$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of both the equations as shown below and for 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 $x\approx -1.911$ and $x\approx 1.911$ .

Therefore, the solution of the equation $\sec x=-3$ in given interval are $x\approx -1.911$ and $x\approx 1.911$ .