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:

Prove that $\sin 10^{\circ}>\dfrac{1}{6}$.

Another heuristic problem solving technique that is so popular and widely use is called "Prove by Contradiction" technique.

Euclid's beautiful proof by contradiction to show there are infinitely many prime numbers is one of the most mind-boggling method to prove something is true that leave us speechless with absolute admiration.