Chapter 5.5, Problem 94E

To determine

To find: The solutions of the given equation in the given interval. Then verify the solutions graphically.

Answer to Problem 94E

The solution of the given equation in the interval $[0,2\pi )$ is $x=0,\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4},\pi ,\frac{5\pi }{4},\frac{3\pi }{2},\frac{7\pi }{4}$ .

Explanation of Solution

Given information:

The following equation:

  $\cos 2x-\cos 6x=0$

The following interval:

  $[0,2\pi )$

Formula used:

The following sum to product formula:

  $\cos u-\cos v=-2\sin \left(\frac{u+v}{2}\right)\sin \left(\frac{u-v}{2}\right)$

Calculation:

The given equation can be solved as follows:

  $\begin{array}{l}\cos 2x-\cos 6x=0\text{ [Given]}\\ -2\sin \left(\frac{6x+2x}{2}\right)\sin \left(\frac{6x-2x}{2}\right)=0\text{ [Sum to product formula]}\\ -2\sin \left(\frac{8x}{2}\right)\sin \left(\frac{4x}{2}\right)=0\text{ [Add and subtract]}\\ -2\sin 4x\sin 2x=0\text{ [Simplify]}\\ \sin 4x\sin 2x=0\text{ [Divide both sides by }-\text{2]}\\ \sin 4x=0\text{ sin}2x=0\text{ [Set each factor to 0]}\\ 4x=\arcsin (0)2x=\arcsin (0)\text{ [Inverse trigonometric function]}\\ 4x=n\pi 2x=n\pi \text{ [Value of inverse]}\\ x=\frac{n\pi }{4}x=\frac{n\pi }{2}\text{ [Simplify]}\\ x=0,\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4}\text{,}\pi \text{,}\frac{5\pi }{4}\text{,}\frac{3\pi }{2}\text{,}\frac{7\pi }{4}\text{ [Solutions in the interval [0,2}\pi \text{)]}\end{array}$

The solution is verified graphically as:

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression $\cos 2x-\cos 6x$ and which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation:

In the interval $[0,2\pi )$ , the value of the equation is $0$ at $x=0,\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4},\pi ,\frac{5\pi }{4},\frac{3\pi }{2},\frac{7\pi }{4}$ . Hence, verified.