Tangent of 165 Degrees

Theorem

$\tan 165 \degrees = \tan \dfrac {11 \pi} {12} = -\paren {2 - \sqrt 3}$

where $\tan$ denotes tangent.


Proof

\(\ds \tan 165 \degrees\) \(=\) \(\ds \map \tan {90 \degrees + 75 \degrees}\)
\(\ds \) \(=\) \(\ds -\cot 75 \degrees\) Tangent of Angle plus Right Angle
\(\ds \) \(=\) \(\ds -\paren {2 - \sqrt 3}\) Cotangent of $75 \degrees$

$\blacksquare$


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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