Chapter 5.5, Problem 96E

To determine

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

Answer to Problem 96E

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:

  ${\sin }^{2}3x-{\sin }^{2}x=0$

The following interval:

  $[0,2\pi )$

Formula used:

The following power reduction formula:

  ${\sin }^{2}u=\frac{1-\cos 2u}{2}$

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:

  $\begin{array}{l}{\sin }^{2}3x-{\sin }^{2}x=0\text{ [Given]}\\ \frac{1-\cos 6x}{2}-\frac{1-\cos 2x}{2}=0\text{ [Power reduction formula]}\\ \frac{1-\cos 6x-1+\cos 2x}{2}=0\text{ [Subtract]}\\ 1-\cos 6x-1+\cos 2x=0\text{ [Multiply both sides by 2]}\\ -\cos 6x+\cos 2x=0\text{ [Add like terms]}\\ \cos 6x-\cos 2x=0\text{ [Reverse signs to make first term positive]}\\ -2\sin \left(\frac{6x+2x}{2}\right)\sin \left(\frac{6x-2x}{2}\right)=0\text{ [sum to product formula]}\\ -2\sin \left(\frac{8x}{2}\right)\sin \left(\frac{4x}{2}\right)=0\text{ [Add and subtract]}\\ -2\sin 4x\sin 2x=0\text{ [Simplify]}\\ \sin 4x\sin 2x=0\text{ [Divide both sides by }-\text{2]}\\ \sin 4x=0\sin 2x=0\text{ [Set each factor to 0]}\\ 4x=\arcsin 02x=\arcsin 0\text{ [Inverse trigonometric function]}\\ 4x=n\pi 2x=n\pi \text{ [Value of inverse]}\\ x=\frac{n\pi }{4}x=\frac{n\pi }{2}\text{ [Simplify]}\\ x=0,\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4}\text{,}\pi \text{,}\frac{5\pi }{4}\text{,}\frac{3\pi }{2}\text{,}\frac{7\pi }{4}\text{ [Solutions in the interval [0,2}\pi \text{)]}\end{array}$

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 ${\sin }^{2}3x-{\sin }^{2}x$ 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.