Sine of 30 Degrees

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$\sin 30 \degrees = \sin \dfrac \pi 6 = \dfrac 1 2$

where $\sin$ denotes the sine function.



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:

$\angle ACD = \dfrac {60 \degrees} 2 = 30 \degrees$

So by definition of sine function:

$\sin \paren {\angle ACD} = \dfrac {r / 2} r = \dfrac 1 2$