Cosecant of 285 Degrees

Theorem

$\csc 285 \degrees = \csc \dfrac {19 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$

where $\csc$ denotes cosecant.


Proof

\(\ds \csc 285 \degrees\) \(=\) \(\ds \map \csc {360 \degrees - 75 \degrees}\)
\(\ds \) \(=\) \(\ds -\csc 75 \degrees\) Cosecant of Conjugate Angle
\(\ds \) \(=\) \(\ds -\paren {\sqrt 6 - \sqrt 2}\) Cosecant 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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