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Is there a number whose cosine is itself? …

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

Is there a number whose cosine is itself? Let $h(x) = \cos x - x$ , with $x$ in radians. Find $h(1)$ and show that $h$ changes sign on $[0, 1]$ . Then halve the interval three times, finding $h(0.75)$ along the way, and give the width of the interval that remains.

Need a reminder? The key ideaShow

If $f$ is continuous on $[a, b]$ and $f(a)$ and $f(b)$ have opposite signs, then $f(c) = 0$ for some $c$ in $(a, b)$ . Halving the interval and keeping the half with the sign change traps the root.

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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