By indicating respectively with $\theta_{\min}$ and $\theta_{\max}$ the minimum angle and the maximum angle of a triangle, we have:
$$ \text{quality}(\theta_{\min},\,\theta_{\max}) := \max\left(\frac{\pi/3-\theta_{\min}}{\pi/3},\,\frac{\theta_{\max}-\pi/3}{2\,\pi/3}\right) $$
that is a function with image the interval $[0,\,1]$ that tries to give an indication about the goodness of a triangle.
In particular, we obtain $0$ in the case of equilateral triangles, $1$ in the case of degenerate triangles.
At this point, I was wondering if it was possible to get the same function, but in terms of $\cos\theta_{\min}$, $\cos\theta_{\max}$.
I tried to modify the function in the following way:
$$ \text{quality}(\theta_{\min},\,\theta_{\max}) := \max\left(\frac{1/2-\cos\theta_{\min}}{1/2},\,\frac{\cos\theta_{\max}-1/2}{-1/2}\right) $$
but obviously the same values are not obtained, the image set is different!
Some advice? Thank you!