Chapter 14.2, Problem 58PPSE

Solution

To identify : The $x-$ intercepts of the graph of the function $y=2{\sin }^4\theta -{\sin }^2\theta$ .

The $x-$ intercepts of the graph of the function are $\frac{\pi }{4}+\frac{\pi n}{2}$ and $0+\pi n$ , where n is any integer.

Given information :

The function is: $y=2{\sin }^4\theta -{\sin }^2\theta$ .

Concept used:

The $x-$ intercept of a graph is where it intersects the $x-$ axis or where $y=0$ .

Explanation :

Given function is $y=2{\sin }^4\theta -{\sin }^2\theta$ .

The graph of the function is shown as below:

  

The $x-$ intercepts are $\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4},\mathrm{....}$

Consider the above graph.

  $\frac{3\pi }{4}-\frac{\pi }{4}=\frac{\pi }{2}$

So, the difference between the intercepts is $\frac{\pi }{2}$ .

Similarly, the $x-$ intercepts are $0,\pi ,2\pi ,3\pi ,\mathrm{....}$

In general form, $2\pi -\pi =\pi$

So, the difference between the intercepts is $\pi$ .

Therefore, the $x-$ intercepts of the graph of the given function are $\frac{\pi }{4}+\frac{\pi n}{2}$ and $0+\pi n$ , where n is any integer.