Law of cotangents

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.

s = (a + b + c)/2

Then the law of cotangents says

\frac{\cot \alpha/2}{s - a} = \frac{\cot \beta/2}{s - b} = \frac{\cot \gamma/2}{s - c} = \frac{1}{r}

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

r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}

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

A = \sqrt{s(s-a)(s-b)(s-c)}

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.”

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