Question

Write the number of points of intersection of the curves $2y=1$ and $y=\cos x,0\le x\le 2\pi$.

Answer 1

Given curves: $2y=1$and $y=\cos x$
Now,
$2y=1\Rightarrow y=\frac{1}{2}$
Also,

$$\begin{gathered} \cos x=y \\ \Rightarrow \cos x=\frac{1}{2} \\ \Rightarrow \cos x=\cos \left(\frac{\mathrm{\pi }}{3}\right)\mathrm{and}\cos x=\cos \left(\frac{4\mathrm{\pi }}{3}\right) \end{gathered}$$

$$\begin{gathered} \Rightarrow x=2n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{3}\mathrm{or}x=2n\mathrm{\pi }\pm \frac{4\mathrm{\pi }}{3} \\ \mathrm{By}\mathrm{putting}n=0,\mathrm{we}\mathrm{get}: \\ x=\frac{\mathrm{\pi }}{3}\mathrm{and}x=\frac{2\mathrm{\pi }}{3} \end{gathered}$$

For the other value of n, the value of x will not satisfy the given condition.

Hence, the number of points of intersection of the curves is two, i.e., $\frac{\mathrm{\pi }}{3}\mathrm{and}\frac{4\mathrm{\pi }}{3}$.