Chapter 1.6, Problem 36E

To determine

To find: The solution of the equation $\cot x=-1$ graphically in the given interval $-\infty <x<\infty$ .

Answer to Problem 36E

The solution of the equation $\cot x=-1$ in given interval is $x=-\frac{\pi }{4}+k\pi$ .

Explanation of Solution

Given information:The given equation is $\cot x=-1$ with the interval $-\infty <x<\infty$ .

Calculation:

The solutions of given equation $\cot x=-1$ are the intersecting points of both the curves $y=\cot x$ and $y=-1$ for $-\infty <x<\infty$ .

To graph a function $y=\cot x$ 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 $y=\cot x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{1}\boxed{÷}\boxed{\text{TAN}}\boxed{\text{X}}\boxed{)}$ . 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 $\boxed{(-)}\boxed{1}$ .

The display will show the equations,

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

Press the $\boxed{\text{WINDOW}}$ key and adjust the window to $\left[-\pi ,\pi \right]$ by $\left[-1.1,1.1\right]$ .

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 $x=-\frac{\pi }{4}$ .

The value of the function $y=\cot x$ repeats its value after every $\pi$ . So, the general solution of the equation $\cot x=-1$ is $x=-\frac{\pi }{4}+k\pi$ , for any integer k .

Therefore, the solution of the equation $\cot x=-1$ in given interval is $x=-\frac{\pi }{4}+k\pi$ .