Sine of Angle plus Straight Angle

Theorem

$\map \sin {x + \pi} = -\sin x$


Proof

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

$\blacksquare$


Also see


Sources

  • 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$
  • 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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