How to find the exact value of tan(pi/3)?

Question

sobrevenilac · Accepted Answer

This fundamental trigonometric identity can be used:
\(\displaystyle{\tan{\theta}}=\frac{{\sin{\theta}}}{{\cos{\theta}}}\)
Here's a reference triangle with our \(\displaystyle\angle\theta\):
Since we know \(\displaystyle{\sin{{\left(\frac{\pi}{{3}}\right)}}}\) is \(\displaystyle\frac{\sqrt{{3}}}{{2}}\) and \(\displaystyle{\cos{{\left(\frac{\pi}{{3}}\right)}}}\) is \(\displaystyle\frac{{1}}{{2}}\), we can use the previously stated identity to figure out the value of \(\displaystyle{\tan{{\left(\frac{\pi}{{3}}\right)}}}\):
\(\displaystyle{\tan{{\left(\frac{\pi}{{3}}\right)}}}=\frac{{\quad{\sin{{\left(\frac{\pi}{{3}}\right)}}}\quad}}{{\cos{{\left(\frac{\pi}{{3}}\right)}}}}\)
\(\displaystyle{\color{white}{{\tan{{\left(\frac{\pi}{{3}}\right)}}}}}=\frac{{\quad\frac{\sqrt{{3}}}{{2}}\quad}}{{\frac{{1}}{{2}}}}\)
\(\displaystyle{\color{white}{{\tan{{\left(\frac{\pi}{{3}}\right)}}}}}=\frac{\sqrt{{3}}}{{2}}\cdot\frac{{2}}{{1}}\)
\(\displaystyle{\color{white}{{\tan{{\left(\frac{\pi}{{3}}\right)}}}}}=\frac{\sqrt{{3}}}{{\color{red}\cancel{{\color{black}{2}}}}}\cdot\frac{{\color{red}\cancel{{\color{black}{2}}}}}{{1}}\)
\(\displaystyle{\color{white}{{\tan{{\left(\frac{\pi}{{3}}\right)}}}}}=\frac{\sqrt{{3}}}{{1}}\cdot\frac{{1}}{{1}}\)
\(\displaystyle{\color{white}{{\tan{{\left(\frac{\pi}{{3}}\right)}}}}}=\frac{\sqrt{{3}}}{{1}}\cdot{1}\)
\(\displaystyle{\color{white}{{\tan{{\left(\frac{\pi}{{3}}\right)}}}}}=\frac{\sqrt{{3}}}{{1}}\)
\(\displaystyle{\color{white}{{\tan{{\left(\frac{\pi}{{3}}\right)}}}}}=\sqrt{{3}}\)
That's the value of \(\displaystyle{\tan{{\left(\frac{\pi}{{3}}\right)}}}\).

How to find the exact value of tan(pi/3)?

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