Solutions of cos x equals cos a/Proof

Theorem

Let $\alpha \in \closedint {-1} 1$ be fixed.

Let:

$(1): \quad \cos x = \cos \alpha$


The solution set of $(1)$ is:

$\set {x \in \R: \forall n \in \Z: x = 2 n \pi \pm \alpha}$


Proof

From Cosine of Supplementary Angle:

$\map \cos {\pi - x} = -\cos x$

and so from Real Cosine Function is Periodic:

\(\ds x\) \(=\) \(\ds 2 n \pi \pm a\)

$\blacksquare$


Sources

  • 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: General solution of $\cos x = \cos \alpha$
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