Find number of real solutions of $y^2+y=\sin(x)$ and $y+y^3=\cos^{-1}(\cos(x))$
My try: I started out by trying extreme values i.e. $x=0$ and $y=0$ which was a possible solution. To find the other solutions:
I tried finding the behaviour of the function but approached a dead end as $\cos^{-1}(\cos(x))$ is a periodic function so I couldn't easily find its derivative. Finally, I used Geogebra to graph the function and it had infinite solutions.
This seems obvious by intuition that the curves will intersect each other at infinite number of points as $\cos^{-1}(\cos(x))$ is periodic ($2\pi$) but can anyone prove mathematically that there are infinite pairs $(x,y)$?
Best Answer
As you know that $\sin x$ and $\cos^{-1}(\cos x)$ are periodic with period $2π$ and that $(x, y) =(0, 0) $ is one of the solutions of the equations, it is evident that $x=2nπ,\ n\in\mathbb Z$ is also the solution of both the equations.
Hence, there are infinite pairs.