\(\displaystyle{\tan{{\left({\frac{{\pi}}{{{6}}}}\right)}}}={\frac{{{\sin{{\left({\frac{{\pi}}{{{6}}}}\right)}}}}}{{{\cos{{\left({\frac{{\pi}}{{{6}}}}\right)}}}}}}={\frac{{{1}}}{{\sqrt{{{3}}}}}}={\frac{{\sqrt{{{3}}}}}{{{3}}}}\)

hope that helped

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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