Question

Write the number of points of intersection of the curves

\[2y = 1\] and \[y = \cos x, 0 \leq x \leq 2\pi\].

 

Answer 1

Given curves: 

\[2y = 1\] and

\[y = \cos x\] 
Now, \[2y = 1 \Rightarrow y = \frac{1}{2}\]
Also,
\[\cos x = y\]
\[ \Rightarrow \cos x = \frac{1}{2}\]
\[ \Rightarrow \cos x = \cos \left( \frac{\pi}{3} \right)\text{ and }\cos x = \cos \left( \frac{4\pi}{3} \right)\]

\[\Rightarrow x = 2n\pi \pm \frac{\pi}{3} \text{ or }x = 2n\pi \pm \frac{4\pi}{3}\]

By putting n = 0, we get: 

\[x = \frac{\pi}{3}\text{ and }x = \frac{2\pi}{3}\]
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.,
\[\frac{\pi}{3}\text{ and }\frac{4\pi}{3}\]