Parametric version of a simple equation

Question

I have a simple relation that I need to plot in a plane. I could do it, but I believe that I don't get the best way. A plane curve is defined implicitely by the following equation : \begin{equation}\tag{1} \cos{\chi} + \sin{\eta} = \sqrt{f} \; \cos{\eta}, \end{equation} where $f$ is a positive constant. I need to plot the curve in the $(\chi, \eta)$ plane. It's easy to invert the relation above (expressing $\chi$ as a function of $\eta$ is simpler but isn't well adapted to the curve) : \begin{equation} \tag{2} \sin{\eta} = g_f(\chi) \equiv \frac{\sqrt{f^2 + f \, \sin^2{\chi}} - \cos{\chi}}{1 + f}, \end{equation} so I could define a parametric version of equation (1) above : \begin{equation} \tag{3} (\chi, \; \arcsin{g_f(\chi)}), \end{equation} This does the job. However, I strongly suspect there's a simpler parametrisation of the same curve, represented by equation (1).

So the question is what is the best or simplest parametrisation $(\chi(s), \; \eta(s))$ of the equation (1) above ?

EDIT : Take note that $\chi$ is only defined on the interval $-\pi$ to $\pi$. Also : $-\frac{\pi}{2} < \eta < \frac{\pi}{2}$.

Answer 1

Your equation can be rearranged to $$ \cos\chi = \sqrt{f}\cos\eta - \sin\eta. \tag{1} $$ Let $\phi$ be an angle (unique up to an added integer multiple of $2\pi$) satisfying $$ \cos\phi = \sqrt{\frac{f}{f + 1}},\qquad \sin\phi = \frac{1}{\sqrt{f + 1}}. $$ By the sum formula for cosine, (1) becomes $$ \cos\chi = \sqrt{f + 1}(\cos\phi \cos\eta - \sin\phi \sin\eta) = \sqrt{f + 1} \cos(\phi + \eta). $$ This can be written $$ \eta = -\phi + \arccos\left(\frac{\cos\chi}{\sqrt{f + 1}}\right). $$

Answer 2

Using Andrew's answer, I could define the following parametrization, which appears to be working great : \begin{align} \big(\, \chi(s), \; \eta(s) \big) = \big(\arccos{s}, \quad \arccos{\frac{s}{\sqrt{1 + f}}} - \arccos{\sqrt{\frac{f}{1 + f}}} \big), \end{align} where $-1 < s < 1$ (or $s = \cos{\chi}$). But I still suspect this could be improved (i.e. simplified). That parametrization isn't much simpler than the expression (2) I gave in my question.