Tangent of 105 Degrees

Theorem

$\tan 105^\circ = \tan \dfrac {7 \pi} {12} = - \left({2 + \sqrt 3}\right)$

where $\tan$ denotes tangent.


Proof

\(\ds \tan 105^\circ\) \(=\) \(\ds \tan \left({90^\circ + 15^\circ}\right)\)
\(\ds \) \(=\) \(\ds - \cot 15^\circ\) Tangent of Angle plus Right Angle
\(\ds \) \(=\) \(\ds - \left({2 + \sqrt 3}\right)\) Cotangent of 15 Degrees

$\blacksquare$


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
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