Cosecant of 240 Degrees

Theorem

$\csc 240 \degrees = \csc \dfrac {4 \pi} 3 = -\dfrac {2 \sqrt 3} 3$

where $\csc$ denotes cosecant.


Proof

\(\ds \csc 240 \degrees\) \(=\) \(\ds \map \csc {360 \degrees - 120 \degrees}\)
\(\ds \) \(=\) \(\ds -\csc 120 \degrees\) Cosecant of Conjugate Angle
\(\ds \) \(=\) \(\ds -\frac {2 \sqrt 3} 3\) Cosecant of $120 \degrees$

$\blacksquare$


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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