Chapter 1.6, Problem 6QR

To determine

To find: Thesolution of the equation $\cos x=-0.4$ graphically in the given interval $0\le x\le 2\pi$ .

Answer to Problem 6QR

The solutions of the equation $\cos x=-0.4$ in the given interval $0\le x\le 2\pi$ are $x\approx 1.982$ and $x\approx 4.301$ .

Explanation of Solution

Given information:

The given equation is $\cos x=-0.4$ and the interval is $0\le x\le 2\pi$ .

Calculation:

The solutions of given equation $\cos x=-0.4$ are the intersecting points of both the curves $y=\cos x$ and $y=-0.4$ for $0\le x\le 2\pi$ .

To graph a function $y=\cos x$ and $y=-0.4$ , 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 $y=\cos x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{COS}}\boxed{\text{X}}\boxed{)}$ . Now press the $\boxed{\text{ENTER}}$ key and enter right hand side of the equation $y=0.6$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{(-)}\boxed{0.4}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\cos \left(\text{X}\right)\\ {\text{Y}}_2=-0.4\end{array}$

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. 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 $x\approx 1.982$ and $x\approx 4.301$ .

Therefore, the solutions of the equation $\cos x=-0.4$ in the given interval $0\le x\le 2\pi$ are $x\approx 1.982$ and $x\approx 4.301$ .