Exercise: error of the sin x ≈ x approximation

For small values, $\sin x\approx x$. Compute the numerical and percentage error of using that approximation at $x=\pi/8$.

1. Analytic and approximate solution:

   $$y=\sin\!\left(\tfrac{\pi}{8}\right)=\frac{\sqrt{2-\sqrt2}}{2},\qquad \hat y=\frac{\pi}{8}$$

2. Numerical error (rounded to six decimals):

   $$\epsilon=\left|\frac{\sqrt{2-\sqrt2}}{2}-\frac{\pi}{8}\right|=0.010016$$