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

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

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Thomas Lynn
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

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.
$$\tan(\frac{\pi}{6})=\approx0.577$$

user_27qwe
Thank u a lot