Tangent of 285 Degrees

Theorem

$\tan 285 \degrees = \tan \dfrac {19 \pi} {12} = -\paren {2 + \sqrt 3}$

where $\tan$ denotes tangent.


Proof

\(\ds \tan 285 \degrees\) \(=\) \(\ds \map \tan {360 \degrees - 75 \degrees}\)
\(\ds \) \(=\) \(\ds -\tan 75 \degrees\) Tangent of Conjugate Angle
\(\ds \) \(=\) \(\ds -\paren {2 + \sqrt 3}\) Tangent 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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