# Trisecting an Angle

- Author:
- Ku, Yin Bon (Albert)

Given any angle, we already know that it can be bisected by Euclidean construction. Then how about trisecting an angle? For some special angles like , we can certainly trisect it because we known how to construct a angle. However, what we really want is a way to trisect

**ANY**angle by Euclidean construction. In this section, we will show that this is**IMPOSSIBLE**- there exists an angle that cannot be trisected! To prove this, we first need to define what is a "**constructible angle**" and its relation to constructible numbers. Then we use some trigonometric identities to derive a cubic equation so as to prove the impossibility using the Main Theorem.## Constructible angles

An angle is said to be

**constructible**if it can be constructed by straightedge and compass only i.e the angle is formed by three constructible points. It is not hard to deduce the following simple but important result:An angle is constructible if and only if is a constructible number.