Cosine of Angle plus Right Angle

Theorem

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


Proof

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

$\blacksquare$


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
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Shifts and periodicity
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Shifts and periodicity
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