Intermediate Value Theorem to Prove Inequality Problem

In my previous post, I talked about how to use calculus method to prove that $\sin 10^{\circ} > \dfrac{1}{6}$ (Never Be Complacent).

Or more precisely, the Intermediate Value Theorem as our weapon to prove it.

We will use the triple angle identity for $\sin 3x=3\sin x-4\sin^3 x$ and the value for $\sin 30^{\circ}=\dfrac{1}{2}$ when $x=10^{\circ}$ in our solution: