91Ó°ÊÓ

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.

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 $(\frac{\mathrm{Ï€}}{3},\frac{1}{2})$.

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

So

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

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

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

Now,

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

and

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

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