Chapter 1.6, Problem 33E

To determine

To find: The solution of the equation $\csc x=2$ graphically in the given interval $0<x<2\pi$ .

Answer to Problem 33E

The solution of the equation $\csc x=2$ in given interval are $x=\frac{\pi }{6}$ and $x=\frac{5\pi }{6}$ .

Explanation of Solution

Given information:The given equation is $\csc x=2$ with the interval $0<x<2\pi$ .

Calculation:

The solutions of given equation $\csc x=2$ are the intersecting points of both the curves $y=\csc x$ and $y=2$ for $0<x<2\pi$ .

To graph a function $y=\csc x$ and $y=2$ , 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 $y=\csc x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{1}}\boxed{÷}\boxed{\text{SIN}}\boxed{\text{X}}\boxed{)}$ . Now, press the $\boxed{\text{ENTER}}$ key and enter right hand side of the equation $y=2$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{2}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=1/\sin \left(\text{X}\right)\\ {\text{Y}}_2=2\end{array}$

Press the $\boxed{\text{WINDOW}}$ key and adjust the window to $\left[0,2\pi \right]$ by $\left[-3,3\right]$ .

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 $x=\frac{\pi }{6}$ and $x=\frac{5\pi }{6}$ .

Therefore, the solution of the equation $\csc x=2$ in given interval are $x=\frac{\pi }{6}$ and $x=\frac{5\pi }{6}$ .