Cosine of 135 Degrees

Theorem

$\cos 135 \degrees = \cos \dfrac {3 \pi} 4 = -\dfrac {\sqrt 2} 2$

where $\cos$ denotes cosine.


Proof

\(\ds \cos 135 \degrees\) \(=\) \(\ds \map \cos {90 \degrees + 45 \degrees}\)
\(\ds \) \(=\) \(\ds -\sin 45 \degrees\) Cosine of Angle plus Right Angle
\(\ds \) \(=\) \(\ds -\frac {\sqrt 2} 2\) Sine of $45 \degrees$

$\blacksquare$


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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