In a recent post I described a method of generating the simplest primitive Pythagorean triple (a,b,c) where one of the angles of the triangle with sides a, b and c is θ° to within some (small) error bound Δθ°.

One of the steps was, given the cosine C of the angle [from step (2)]:

(3) Calculate the Farey ratio approximant …

…

Now, that function

seemed semi-familiar, resembling functions that occur in trigonometric or hyperbolic identities.

An example is:

A little further investigation, and reading around, including the Wikipedia articles on trigonometric identities, and in particular on those of the tangent half-angle, revealed that the *Farey ratio approximant* does in fact correspond directly to a simple trigonometric function of the angle:

The slightly simplified method follows.