Chapter 5.1, Problem 63E

Solution

To calculate: The solutions of the trigonometric equation $\cos x=0.37$ .

All the solutions of the equation $\cos x=0.37$ are $x=\left\{1.19+2n\pi |n=0,\pm 1,\pm 2,\dots \right\}$ and $x=\left\{5.09+2n\pi |n=0,\pm 1,\pm 2,\dots \right\}$ .

Given information: The given equation is $\cos x=0.37$ .

Calculation:

To graph a function $\cos x=0.37$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key. Enter left side of the equation $\cos x=0.37$ after the symbol ${\text{Y}}_1$ and enter the keystrokes $\boxed{\text{COS}}\boxed{\text{X}}$ . Enter right side of the equation $\cos x=0.37$ after the symbol ${\text{Y}}_2$ and enter the keystrokes $\boxed{\text{0}}\boxed{\text{.}}\boxed{\text{3}}\boxed{\text{7}}$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph the equation in the interval $\left[0,2\pi \right)$ as shown in Figure (1).

  

Figure (1)

The intersection of the graphs gives two solutions of the equation $\cos x=0.37$ in the interval $\left[0,2\pi \right)$ . The two values are $x=1.19$ and $x=5.09$ . Use the period of cosine $2\pi$ to find the complete solution set $x=\left\{1.19+2n\pi |n=0,\pm 1,\pm 2,\dots \right\}$ and $x=\left\{5.09+2n\pi |n=0,\pm 1,\pm 2,\dots \right\}$ .

Thus, all the solutions of the equation $\cos x=0.37$ are $x=\left\{1.19+2n\pi |n=0,\pm 1,\pm 2,\dots \right\}$ and $x=\left\{5.09+2n\pi |n=0,\pm 1,\pm 2,\dots \right\}$ .