Sine of Angle plus Three Right Angles

Theorem

$\map \sin {x + \dfrac {3 \pi} 2} = -\cos x$


Proof

\(\ds \map \sin {x + \frac {3 \pi} 2}\) \(=\) \(\ds \sin x \cos \frac {3 \pi} 2 + \cos x \sin \frac {3 \pi} 2\) Sine of Sum
\(\ds \) \(=\) \(\ds \sin x \cdot 0 + \cos x \cdot \paren {-1}\) Cosine of Three Right Angles and Sine of Three Right Angles
\(\ds \) \(=\) \(\ds -\cos x\)

$\blacksquare$


Also see


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
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