# How do you evaluate \sin(\frac{\pi}{6}) ?

How do you evaluate $$\displaystyle{\sin{{\left({\frac{{\pi}}{{{6}}}}\right)}}}$$ ?

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Tiefdruckot
Explanation:
Start with an equilateral triangle of side 2. The interior angle at each vertex must be $$\displaystyle{\frac{{\pi}}{{{3}}}}$$ since 6 such angles make up a complete $$\displaystyle{2}\pi$$ circle.
Then bisect the triangle through a vertex and the middle of the opposite side, dividing it into two right angled triangles.
These will have sides of length 2,1 and $$\displaystyle\sqrt{{{2}^{{2}}-{1}^{{2}}}}=\sqrt{{{3}}}$$. The interior angles of each right angled triangle are $$\displaystyle{\frac{{\pi}}{{{3}}}},\ {\frac{{\pi}}{{{6}}}}$$ and $$\displaystyle{\frac{{\pi}}{{{2}}}}$$ with the $$\displaystyle{\frac{{\pi}}{{{6}}}}$$ coming from the fact that we have bicested one of the $$\displaystyle{\frac{{\pi}}{{{3}}}}$$ angles.
Then:
$$\displaystyle{\sin{{\left({\frac{{\pi}}{{{6}}}}\right)}}}={\frac{{\text{opposite}}}{{\text{hypotenuse}}}}={\frac{{12}}{}}$$
###### Not exactly what youâ€™re looking for?
user_27qwe

The exact value of $$\sin(\frac{\pi}{6})\ \text{is}\ \frac12$$
The result can be shown in multiple forms
Exact Form:$$\ \frac12$$
Decimal Form:
0.5