We obtain a single equation to parametrization of phase's angle $p$ for the periodic cosine function when the typical condition $\cos(x+p)=cos x$ is satisfied. Without losing generality, we will restrict the discussion to the case $0\leq x \leq 2\pi$ . We starting setting $\cos x=t$ , so that $-1\leq t\leq 1$ is the interval of definition for the real parameter $t$ . Next we expand out \begin{equation*} \cos(x+p)=\cos x\cos p-\sin x\sin p, \end{equation*}where we introduce $t$ and apply the assumed condition to get \begin{equation*} t\cos p-\sqrt{1-t^2}\sin p=t. \end{equation*}This leads to a quadratic equation in $\cos p$ on terms of parameter $t$ , i.e. \begin{equation*} \cos^2p(t)-2t^2\cos p(t)+(2t^2-1)=0. \end{equation*}Solving, \begin{equation*} \cos p(t)=t^2\pm (t^2-1). \end{equation*}One root has to do with the trivial $\cos(x+2\pi)=\cos x$ , but we are interested on the nontrivial one \begin{equation} \cos p(t)=2t^2-1, \qquad -1<t<1. \end{equation}Locus of (1) is the below shown parabola.

unit=2cm

Inverse function of (1) is the wanted parametrization $p(t)$ , with codomain $[0,\pi]$ and locus the ``arabian dome'' below shown.¹

unit=2cm

Footnotes

...¹

Graphics by pahio, with permission.