I was given a problem which ended up solving the following equation for $x$, with $r \in \mathbb{R}^+$ : $$ \forall x \in [0,2r] , \cos^{-1} \left(\frac{x}{2r}\right)(4r^2-2x^2)+x\sqrt{4r^2-x^2}-\pi r^2=0 $$

Having only a high-school level, I don't know how to solve this. I guess squaring everything would help, but I'd still be stuck with the $\cos^{-1} \left(\frac{x}{2r}\right)$.
Is it possible or do I have to approximate the answer?

  • $\begingroup$ You might also give us the context for this problem. There may be another way to approach the problem. $\endgroup$ – Simon S Nov 4 '15 at 15:53


First, start changing variable, say $x=2ry$ and simplify ($r$ should disappear from the equation).

Second, there is no analytical solution to equations which mix polynomial and trigonometric terms (this is already the case for $x=\cos(x)$) and only numerical methods would solve the problem.

Look at the plot of the new function, guess an approximate value of the solution. Now, you are ready for using a root-finding method (Newton would probably be the simplest).

  • $\begingroup$ $r$ is a constant so there's no need to simplify, I should have precised it. The second paragraph is the answer I looked for, thanks. $\endgroup$ – Barahir Nov 5 '15 at 17:52
  • $\begingroup$ You are welcome ! $\endgroup$ – Claude Leibovici Nov 5 '15 at 18:42

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