The previous post commented that the law of tangents is much less familiar than the laws of sines and cosines. The law of **cotangents** is even more obscure. If you ask Google’s Ngram viewer to plot occurrences of “law of cotangents” over time, it will return “Ngrams not found: law of cotangents.”

What is this law of cotangents?

As in the previous post, let *a*, *b*, and *c* be the lengths of the sides of a triangle and let α, β, and γ be the angles opposite *a*, *b*, and *c* respectively. Furthermore, let *s* be the semiperimeter, half the perimeter of the triangle.

Then the law of cotangents says

Here *r* is the radius of the incircle. It turns out *r* is given by

This equation for the radius of the inscribed circle may remind you of Heron’s formula for the area of the triangle:

These are related: you can quickly prove Huron’s formula using the law of cotangents.

Are there any more law of <insert trig function>? Not that I know of. Of the six basic trig functions, the only two we’re missing are secant and cosecant, and as far as I can tell there is no “law of secants” or “law of cosecants.”

I demand a law of cohavercosines!