Page 19 - MATINF Nr. 13-14

A Generalization of the Law of Cotangents 19

r r
From (3), tan(β/2) = and tan(γ/2) = . Therefore,
s − c s − a
Å ã
1 1 2s − (a + c)
r + r
α 0 s − c s − a (s − a)(s − c)
cot = =
2 r 2 r 2
1 − 1 −
(s − a)(s − c) (s − a)(s − c)
b
r
(s − a)(s − c)
= (since 2s = a + b + c and by (2))
b
s
rs rs s − b
= = = .
2
(s − a)(s − c) r s r
s − b

Hence
0
cot(α /2) cot(β/2) cot(γ/2) 1
= = = .
s − b s − c s − a r
0
0
0
Relabeling a := |BC|, b := |CA|, c := |AB| gives
0
cot(α /2) cot(β/2) cot(γ/2) 1
= = = ,
s − a 0 s − b 0 s − c 0 r
which is precisely the classical law of cotangents for 4ABC.

References

[1] Casey, J. (1888). A Treatise on Plane Trigonometry, Containing an Account of Hyperbolic
Functions; With Numerous Examples. Dublin, Ireland: Hodges, Figgis & Co., pp. 185–186.
Available at https://archive.org/details/treatiseonplanet00caseuoft/page/186/
mode/2up.

[2] Euclid (1956). The Thirteen Books of Euclid’s Elements, Vol. 2, 2nd ed. (T. L. Heath, trans.).
New York, NY: Dover. Book III, Prop. 32.

[3] Law of cotangents. (2024). Wikipedia.Available at
https://en.wikipedia.org/w/index.php?title=Law of cotangents&oldid=1284736
895.

[4] Newman, J. R. (1976). The Universal Encyclopedia of Mathematics. London, UK: Pan Books,
p. 530. (English edition originally published 1964 by George Allen & Unwin, London.)
Available at
https://ia801503.us.archive.org/10/items/in.ernet.dli.2015.96640/2015.96640.
The-Universal-Encyclopedia-Of-Mathematics text.pdf.