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

Pamela Meyer 2021-12-31 Answered
How do you evaluate \(\displaystyle{\tan{{\left({\frac{{\pi}}{{{6}}}}\right)}}}\)?

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Expert Answer

Thomas Lynn
Answered 2022-01-01 Author has 1026 answers
Explanation:
\(\displaystyle{\tan{{\left({\frac{{\pi}}{{{6}}}}\right)}}}={\frac{{{\sin{{\left({\frac{{\pi}}{{{6}}}}\right)}}}}}{{{\cos{{\left({\frac{{\pi}}{{{6}}}}\right)}}}}}}={\frac{{{1}}}{{\sqrt{{{3}}}}}}={\frac{{\sqrt{{{3}}}}}{{{3}}}}\)
hope that helped
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karton
Answered 2022-01-08 Author has 8659 answers

Using the identity
\(\tan=\frac{\sin}{\cos}\)
and
\(\sin(\frac{\pi}{6})=\sin(30)=\frac{1}{2}\)
\(\cos(\frac{\pi}{6})=\cos(30)=\frac{\sqrt{3}}{2}\)
then
\(\tan(\frac{\pi}{6})=\frac{\frac12}{\frac{\sqrt{3}}{2}}\)
You should know that dividing by one number is the same as multiplying by its reciprocal, so
\(\frac{\frac12}{\frac{\sqrt{3}}{2}}=\frac12\cdot\frac{2}{\sqrt{3}}\)
Cancelling the 2's and rationalising the denominator,
\(\frac12\cdot\frac{2}{\sqrt{3}}=\frac{1}{\sqrt{3}}\)
\(\frac{1}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{3}\)
Therefore,
\(\tan(\frac{\pi}{6})=\frac{\sqrt{3}}{3}\)
Using a calculator.
\(\tan(\frac{\pi}{6})=\approx0.577\)

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user_27qwe
Answered 2022-01-08 Author has 9558 answers
Thank u a lot
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