Sine of 30 Degrees

Theorem

 * $\sin 30^\circ = \sin \dfrac \pi 6 = \dfrac 1 2$

where $\sin$ denotes the sine function.

Proof

 * Sine30.png

Let $\triangle ABC$ be an equilateral triangle of side $r$.

By definition, each angle of $\triangle ABC$ is equal.

From Sum of Angles of Triangle Equals Two Right Angles it follows that each angle measures $60^\circ$.

Let $CD$ be a perpendicular dropped from $C$ to $AB$ at $D$.

Then $AD = \dfrac r 2$ while:
 * $\angle ACD = \dfrac {60^\circ} 2 = 30^\circ$

So by definition of sine function:
 * $\sin \left({\angle ACD}\right) = \dfrac {r / 2} r = \dfrac 1 2$