Solutions of cos x equals cos a

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:

\(\ds x\)

\(=\)

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

$\blacksquare$

Examples

Solutions of $\cos 2 x = \sin 3 x$

The equation

$\cos 2 x = \sin 3 x$

has the general solution:

$x = \paren {4 n + 1} \dfrac \pi {10}$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(35)$