Finding intersection point of curves $y=\cos x$ and $y = \frac{3x}{2\pi}$ with $x \geq 0$

Question

I just need to find the intersection point of those two curves greater than zero on the $x$-axis. I really only need $x$ now that I think about it. I will use this information to find the area enclosed by the curves.

I completely forget how to go about this, so any help is greatly appreciated. Thank you.

Answer 1

They intersect at $$x=\pi/3$$ because $$\cos (\pi/3)=1/2$$

To prove uniqueness, define $$f(x)=\frac{3}{2\pi}x-\cos x$$ and notice that $$f'(x)=\frac{3}{2\pi}+\sin x$$

Now for $0\le x \le \pi$, $\sin x \ge 0$, and so $f'(x)>0$.

For $x>\pi$, $f(x)>0$ because $|\cos x| \le 1$.