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 [math]90^{\circ}[/math], we can certainly trisect it because we known how to construct a [math]30^{\circ}[/math] angle.  However, what we really want is a way to trisect [b]ANY[/b] angle by Euclidean construction.  In this section, we will show that this is [b]IMPOSSIBLE[/b] - there exists an angle that cannot be trisected!  [br][br]To prove this, we first need to define what is a "[b][color=#0000ff]constructible angle[/color][/b]" 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.

An angle is said to be [b][color=#0000ff]constructible[/color][/b] if it can be constructed by straightedge and compass only i.e the angle is formed by three constructible points.[br][br]It is not hard to deduce the following simple but important result:[br][br][b][i][color=#ff7700][center]An angle [math]\theta[/math] is constructible if and only if [math]\cos(\theta)[/math] is a constructible number.[/center][/color][/i][/b]