Curvilinear Coordinate with coordinate curves that look like cosine in 2D

Question

There are curvilinear coordinates with coordinate curves that look like rays and circles, i.e. polar coordinates, and there are curvilinear coordinates with coordinate curves that look like parabolas, i.e. parabolic coordinates. Can we write down a curvilinear coordinate with one coordinate curve that looks cosine and another coordinate curve being orthogonal to it? Specifically, given a family of curves $$x = -\cos(y)+c,$$ with real number $c$ parametrizing this family of curves, can we find another family curves orthogonal to it and together form a curvilinear coordinate? Any help will be appreciated.

Answer 1

From your family of curves

$$x+\cos y = c$$

implicitly differentiate both sides

$$1 - \sin y \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = \frac{1}{\sin y}$$

then take the negative reciprocal of the quantity to find a new differential equation to solve

$$\frac{dy}{dx} = -\sin y \implies |\sin y| = \operatorname{sech}(x+c)$$

With a little tweaking we find the precise family

$$x + c = \operatorname{sgn}(\cos y)\operatorname{arsech}|\sin y|$$

This is your orthogonal family of curves.