Question

Find the number of the points of intersection of the curves, y = cosx and 2y =1 in the interval $0\le x\le 2\pi$.

Answer 1

Hint: Substitute the value of y from 2y = 1 in the equation y = cosx. Hence form a trigonometric equation. Solve the trigonometric equation using the fact that if cosx = cosy, then $x=2n\pi \pm y$. Hence find the number of solutions of the equation and hence the number of points of intersection of the curves. Alternatively, plot the graph of cosx and draw a line parallel to x -axis at a point $\dfrac{1}{2}$ above it. Find the number of points at which that line intersects the graph of cosx and hence find the number of points of intersection of the curves.

Complete Step-by-step answer:
We have 2y = 1
Hence $y=\dfrac{1}{2}\text{ (i)}$
Also, we have $y=\cos x$
Substituting the value of y from equation (i), we get
$\cos x=\dfrac{1}{2}$.
Now we know that $\cos \left( \dfrac{\pi }{3} \right)=\dfrac{1}{2}$
Hence we have $\cos x=\cos \left( \dfrac{\pi }{3} \right)$.
We know that the solution of the equation $\cos x=\cos y$ is given by $x=2n\pi \pm y,n\in \mathbb{Z}$
Hence we have
$x=2n\pi \pm \dfrac{\pi }{3}$
Put n = 0
We get $x=\pm \dfrac{\pi }{3}$
Since$x\ge 0$, we get $x=\dfrac{\pi }{3}$
Put n = 1
We get $x=2\pi \pm \dfrac{\pi }{3}=\dfrac{5\pi }{3},\dfrac{7\pi }{3}$
Since $x\le 2\pi$, we get
$x=\dfrac{5\pi }{3}$
Hence the number of points of intersection is 2.

Note: The plots of y = cosx (Red) and 2y = cosx (Black) are shown below:

As is clear from the graph the number of points of intersection of the curves = 2 in the interval $\left[ A=0,B=2\pi \right]$ which are C and D.