Does the functional square root of the cosine admit a vector-based interpretation?

Question

In linear algebra, the cosine of the angle between two vectors $a$ and $b$ is defined as $$\cos(a,b) = \frac{\langle a, b \rangle}{||a||\cdot||b||} .$$ The functional square root of the cosine has at various times been studied by mathematicians. It is the function $f(\cdot)$ such that $$f(f(x)) = \cos(x). $$ See for instance this MO question. I wonder whether it is possible to have a vector-based interpretation of this functional square root of the cosine, similar to the cosine itself as defined above.