Cosecant of 345 Degrees

Theorem

$\csc 345 \degrees = \csc \dfrac {23 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$

where $\csc$ denotes cosecant.

Proof

\(\ds \csc 345 \degrees\)

\(=\)

\(\ds \map \csc {360 \degrees - 15 \degrees}\)

\(\ds \)

\(=\)

\(\ds -\csc 15 \degrees\)

Cosecant of Conjugate Angle

\(\ds \)

\(=\)

\(\ds -\paren {\sqrt 6 + \sqrt 2}\)

Cosecant 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