Example 6 Let $a, b, c, x, y, z \in \mathbf{R}_{+}$, satisfying:
$$
c y + b z = a, \quad a z + c x = b, \quad b x + a y = c.
$$

Try to find the minimum value of the function
$$
f(x, y, z) = \frac{x^{2}}{1+x} + \frac{y^{2}}{1+y} + \frac{z^{2}}{1+z}.
$$

[Analysis] On the surface, this problem is about an algebraic function, but it is actually a problem of finding the maximum value of a trigonometric function. The three equations given are the projection theorems of a triangle, and $x, y, z$ are used to hide $\cos A, \cos B, \cos C$ respectively.