Q. 97 - Precalculus Enhanced with Graphing Utilities

Question

The graphs of the given pairs of functions intersect infinitely many times. Find four of these points of intersection.  $$\begin{gathered} y=\cos x \\ y=\frac{1}{2} \end{gathered}$$

Step-by-Step Solution

Answer

The four points of intersection are $(\frac{\mathrm{\pi }}{3},\frac{1}{2}),(\frac{5\mathrm{\pi }}{3},\frac{1}{2}),(-\frac{\mathrm{\pi }}{3},\frac{1}{2}),(-\frac{5\mathrm{\pi }}{3},\frac{1}{2})$.

Step 1. Given Information

We are given two equations $y=\cos x$ and $y=\frac{1}{2}$.

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 cosine 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 $(\frac{\mathrm{\pi }}{3},\frac{1}{2})$.

Step 2. Find the second point

As the function is cyclic, so $\cos x=\cos (2\mathrm{\pi }-\mathrm{x})$.

So 

$$\begin{gathered} \cos \left(\frac{\mathrm{\pi }}{3}\right)=\cos (2\mathrm{\pi }-\frac{\mathrm{\pi }}{3}) \\ \cos \left(\frac{\mathrm{\pi }}{3}\right)=\cos \left(\frac{5\mathrm{\pi }}{3}\right) \end{gathered}$$

Thus the second point will be $(\frac{5\mathrm{\pi }}{3},\frac{1}{2})$

Step 4. Find the third and fourth point

As cosine function is even. So $\cos x=\cos (-x)$

Now,

$\cos \left(\frac{\mathrm{\pi }}{3}\right)=\cos (-\frac{\mathrm{\pi }}{3})$

and

$\cos \left(\frac{5\mathrm{\pi }}{3}\right)=\cos (-\frac{5\mathrm{\pi }}{3})$

So the third and fourth points are $(-\frac{\mathrm{\pi }}{3},\frac{1}{2}),(-\frac{5\mathrm{\pi }}{3},\frac{1}{2})$