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

Patricia Crane 2021-12-24 Answered
How do you evaluate \(\displaystyle{\sin{{\left({\frac{{\pi}}{{{6}}}}\right)}}}\) ?

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Expert Answer

Tiefdruckot
Answered 2021-12-25 Author has 2127 answers
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:
\(\displaystyle{\sin{{\left({\frac{{\pi}}{{{6}}}}\right)}}}={\frac{{\text{opposite}}}{{\text{hypotenuse}}}}={\frac{{12}}{}}\)
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user_27qwe
Answered 2021-12-30 Author has 9557 answers

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

0

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