Chapter 1.6, Problem 35E

To determine

To find: The solution of the equation $\sin x=-0.5$ graphically in the given interval $-\infty <x<\infty$ .

Answer to Problem 35E

The solution of the equation $\sin x=-0.5$ in given interval are $x=\frac{7\pi }{6}+2k\pi$ and $x=\frac{11\pi }{6}+2k\pi$ .

Explanation of Solution

Given information:

The given equation is $\sin x=-0.5$ with the interval $-\infty <x<\infty$ .

Calculation:

The solutions of given equation $\sin x=-0.5$ are the intersecting points of both the curves $y=\sin x$ and $y=-0.5$ for $-\infty <x<\infty$ .

To graph a function $y=\sin x$ and $y=-0.5$ , 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 $y=\sin x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{SIN}}\boxed{\text{X}}\boxed{)}$ . Now press the $\boxed{\text{ENTER}}$ key and enter right hand side of the equation $y=-0.5$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{(-)}\boxed{0.5}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\sin \left(\text{X}\right)\\ {\text{Y}}_2=-0.5\end{array}$

Press the $\boxed{\text{WINDOW}}$ key and adjust the window to $\left[0,2\pi \right]$ 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 two points $x=\frac{7\pi }{6}$ and $x=\frac{11\pi }{6}$ . The value of the function $y=\sin x$ repeats its value after every $2\pi$ . So, the general solution of the equation $\sin x=-0.5$ is $x=\frac{7\pi }{6}+2k\pi$ and $x=\frac{11\pi }{6}+2k\pi$ .

Therefore, the solution of the equation $\sin x=-0.5$ in given interval are $x=\frac{7\pi }{6}+2k\pi$ and $x=\frac{11\pi }{6}+2k\pi$ .