I was working on an approximation for the sine function, in which I needed to calculate the maximum error to work on a compensation polynomial. My approximation was this:
$f(x) = \frac {4} {\pi^2} x (\pi - |x|)$
Then, obviously the error is found by substracting that from $\sin(x)$. In order to find the x-coordinate of the maximum error I took the derivative and and set it equal to zero.
$\begin{align*} \mathrm{err}(x) &= \sin(x) -f(x)\\ \mathrm{err}'(x) &= \cos(x) + \frac {8} {\pi^2}|x| - \frac 4 {\pi} \\ \mathrm{err}'(x) &= 0 \rightarrow \cos(x) = \frac 4 \pi - \frac 8 {\pi^2}|x| \end{align*}$
But I'm only a highschool student and I have no knowledge of the maths required to solve that last equation, if possible. How would one go into solving that last equation exactly?
A small image:
For my particular purpose it would suffice by calculating the value numerically and continue, but just purely for interest I'd like to know the exact value. Not homework.