How to trace the graphic of $\cos(x) + \cosh(y) = k$?

Question

Is there some systematic way to trace the graphic of

$\cos(x) + \cosh(y) = k$

given a fixed value for $k$?

Suppose $k = 1$: if I choose empirically $y = 1.2$, I know that should be

$\cos(x) = - 0.2$

and so I could find all the infinite values of $x$ which verify the equation. But is this a correct way? How can I proceed, given such an implicit equation?

Answer 1

It's easier to graph $y$ in terms of $x$.

Let $z=e^y$ and note that $\cosh y = \frac{1}2(z+z^{-1})$. So $2\cos x + z+z^{-1} = 2k$. Multiplying by $z$ and rearranging, you get:

$$z^2 +2(\cos x-k)z+1 = 0$$

So $z=(k-\cos x)\pm \sqrt{(k-\cos x)^2-1}$

So $$\begin{align}y&=\log\left(k-\cos x\pm \sqrt{(k-\cos x)^2-1}\right) \\ &=\pm \log\left(k-\cos x + \sqrt{(k-\cos x)^2-1}\right) \end{align}$$