Cosine of Conjugate Angle

Theorem

$\map \cos {2 \pi - \theta} = \cos \theta$

where $\cos$ denotes cosine.


That is, the cosine of an angle equals its conjugate.


Proof

\(\ds \map \cos {2 \pi - \theta}\) \(=\) \(\ds \map \cos {2 \pi} \cos \theta + \map \sin {2 \pi} \sin \theta\) Cosine of Difference
\(\ds \) \(=\) \(\ds 1 \times \cos \theta + 0 \times \sin \theta\) Cosine of Full Angle and Sine of Full Angle
\(\ds \) \(=\) \(\ds \cos \theta\)

$\blacksquare$


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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