Sine of 225 Degrees

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Theorem

$\sin 225 \degrees = \sin \dfrac {5 \pi} 4 = -\dfrac {\sqrt 2} 2$

where $\sin$ denotes the sine function.


Proof

\(\ds \sin 225 \degrees\) \(=\) \(\ds \map \sin {360 \degrees - 135 \degrees}\)
\(\ds \) \(=\) \(\ds -\sin 135 \degrees\) Sine of Conjugate Angle
\(\ds \) \(=\) \(\ds -\frac {\sqrt 2} 2\) Sine of $135 \degrees$

$\blacksquare$


Sources