Cosine of Angle plus Full Angle

Theorem

$\map \cos {x + 2 \pi} = \cos x$


Corollary

Let $n \in \Z$ be an integer.

Then:

$\map \cos {x + 2 n \pi} = \cos x$


Proof

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

$\blacksquare$


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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