Tangent of 165 Degrees

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Theorem

$\tan 165^\circ = \tan \dfrac {11 \pi} {12} = - \left({2 - \sqrt 3}\right)$

where $\tan$ denotes tangent.


Proof

\(\displaystyle \tan 165^\circ\) \(=\) \(\displaystyle \tan \left({90^\circ + 75^\circ}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle - \cot 75^\circ\) Tangent of Angle plus Right Angle
\(\displaystyle \) \(=\) \(\displaystyle - \left({2 - \sqrt 3}\right)\) Cotangent of $75^\circ$

$\blacksquare$


Sources