I managed to solve this question but I did it in a way that just isn't good enough. I think the most sensible way to approach the calculation of $I_{n+1}$ is to evaluate the signs of $f(a_n), f(b_n), f(c_n)$, eg for $I_1$ I calculated $a_0 = \frac{\pi}{2} > 0$, $b_0 = \pi > 0$ and $c_0 = \frac{3\pi}{4}$, setting $f(c_0) = \frac{2}{\sqrt{2}} - \frac{3\pi}{4}<0$, $f(a_0) = 2 - \frac{\pi}{2} > 0$, $f(b_0) = -\pi < 0$ setting $I_1 = [\frac{\pi}{2}, \frac{3\pi}{4}]$.
These inequalities seem obvious to me. For $f(c_0) = \frac{2}{\sqrt{2}} - \frac{3\pi}{4}<0$, this is obvious to me since $\frac{2}{\sqrt{2}} < 2$ and $\frac{3\pi}{4} > \frac{9}{4} > 2$. But this logic isn't extendable in the later parts of the question.
Later I needed to calculate $I_2$ and had the equation for $f(c_0)$ as $f(c_0) = 2\sin{(\frac{5\pi}{8}}) - \frac{5\pi}{8}$. Now I need to determine whether this is greater than or less than zero to zero in order to determine what $I_{n+1}$ should be.
I didn't know how to do this using the given approximations and everything I tried was too hand-wavey and stupid. I ended up having to make educated guesses (I was able to tell it was less than zero, and I also managed to get $I_3$ correct with educated guesswork).
How can I provide a solid argument to show whether expressions like these are greater than or less than zero?
A calculator is not allowed in this question.
