Question

The graphs of the given pairs of functions intersect infinitely many times. Find four of these points of intersection.

$$\begin{gathered} y=2\sin x \\ y=-2 \end{gathered}$$

Short answer

The four points of intersection are $\left(-\frac{\mathrm{\pi }}{2},-2\right),\left(\frac{3\mathrm{\pi }}{2},-2\right),\left(\frac{7\mathrm{\pi }}{2},-2\right),\left(\frac{11\mathrm{\pi }}{2},-2\right)$.

Step-by-step solution

Step 1. Given Information  

We are given two equations $y=2\sin x$and $y=-2$.

We need to find any four points of intersection of the two graphs.

We will graph the two equations and then find the first point of intersection and then using the properties of sine function we will find the other three points.

Step 2. Graph the functions 

The graph of the two equations is given as

It can be seen that the line and the sinusoidal curve intersect each other at infinitely many points and the first point of intersection is $(\frac{3\mathrm{\pi }}{2},-2)$.

Step 3. Find the other points 

The sine function is a cyclic function. So

$$\begin{gathered} 2\sin x=2\sin (\mathrm{\pi }-\mathrm{x}) \\ 2\mathrm{sinx}=2s\mathrm{in}(2\mathrm{\pi }+\mathrm{x}) \\ 2\mathrm{sinx}=2\sin (4\mathrm{\pi }+\mathrm{x}) \end{gathered}$$

So next intersection points are

$$\begin{gathered} x=\mathrm{\pi }-\frac{3\mathrm{\pi }}{2} \\ \mathrm{x}=-\frac{\mathrm{\pi }}{2} \end{gathered}$$

or

role="math" $$\begin{gathered} x=2\mathrm{\pi }+\frac{3\mathrm{\pi }}{2} \\ \mathrm{x}=\frac{7\mathrm{\pi }}{2} \end{gathered}$$

or

$$\begin{gathered} x=4\mathrm{\pi }+\frac{3\mathrm{\pi }}{2} \\ \mathrm{x}=\frac{11\mathrm{\pi }}{2} \end{gathered}$$

So the intersecting points are$\left(-\frac{\mathrm{\pi }}{2},-2\right),\left(\frac{3\mathrm{\pi }}{2},-2\right),\left(\frac{7\mathrm{\pi }}{2},-2\right),\left(\frac{11\mathrm{\pi }}{2},-2\right)$.