I was told to find the "bounds on the error" in a problem in Spivak's Calculus, but I have no idea what that means.
Use the third degree Taylor polynomial of $\cos$ at $0$ to show that the solutions of $x^2=\cos x$ are approximately $\pm\sqrt{2/3}$, and find bounds on the error.
I recognize that this reduces to simply $1 - x^2/2! + R(x) = x^2$ and that $R(x)$ is the error. So, um, if we don't have the $R(x)$, since it's probably small, we get the needed approximate solution. Am I on the right track? What do I need to do to make this formal?