Cosecant of 315 Degrees

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Theorem

$\csc 315^\circ = \csc \dfrac {7 \pi} 4 = -\sqrt 2$

where $\csc$ denotes cosecant.


Proof

\(\ds \csc 315^\circ\) \(=\) \(\ds \csc \left({360^\circ - 45^\circ}\right)\)
\(\ds \) \(=\) \(\ds -\csc 45^\circ\) Cosecant of Conjugate Angle
\(\ds \) \(=\) \(\ds -\sqrt 2\) Cosecant of 45 Degrees

$\blacksquare$


Sources