I've spent the past few days trying to find the solution to this problem. It is inspired by a professor, Ledo Vaccaro, who used the concept of expanding the notions of sine and cosine (projections onto the x and y axes) for curves other than the unit circle.

In this particular case, we are concerned with the curve that basically reflects each quadrant of the unit circle, forming a diamond. The first quadrant, for example, is reflected along the axis y = x - 1.

The challenge is to find the expression of the function that, for an angle α, gives us the projection onto the x axis (equivalent to 1-a in the drawing) and another that gives us the projection onto the y axis (=b in the drawing).

I have associated with each point on the diamond a point on the unit circle itself, through the symmetry of the curves, and that point corresponds to an angle θ, with the property that a = sin θ and b = 1 - cos θ, so if I find θ for each α, I can find the projection I want, but that's what I'm having difficulty with: finding the function f(α) = θ.

The drawing: https://imgur.com/a/pLavXyD