The following curious trigonometric identity was discovered by me around 2005 while applying the sine rule to the cosine rule. It seemed so surprising to me at first that I felt compelled to use Sketchpad to quickly check that I had not accidentally made a logical error somewhere in my derivation. While the result was completely new to me, I later found out it is not new or original (see comments further down). I also recall mentioning it some years ago in one of my Math e-Newsletters, but it doesn't appear to be well-known. The result is a sort of 'angle version' of the familiar cosine identity/rule: BC2 = AB2 + AC2 - 2AB.AC.cos A.
Cosine-Sine Angle Rule
In any triangle ABC, the following general identity holds: (sin A)2 = (sin B)2 + (sin C)2 - 2 sin B sin C cos A.
1) Drag any of A, B or C to explore the generality of the result.
2) What does the identity reduce to when ∠A = 90o?
(Note that your deduced result in 2) gives a sort of 'angle version' of the theorem of Pythagoras, which in turn is equivalent to the general identity sin2 B + cos2 B = 1)
Cosine-Sine Angle Rule
Challenge
3) Can you explain why (prove that) this cosine-sine angle rule is true? Can you deduce (prove) it in more than one way?
4) If necessary, click on the Show Hint button for a hint that should help you.
Some Solutions
a) Here is the original solution I produced in 2005: Derivation of Result.
b) When I posted the problem at the Facebook Group Romantics of Geometry in May 2025, the following different proofs were also given: Other Solutions.
Some Further Comments
5) The trigonometric version of Ceva's theorem also replaces 'sides' by 'sines of angles' much like in the result above.
6) If we construct triangles on the sides of a triangle, we get a similar-looking formula to that of Ceva in terms of ratios of sides or ratios of sines of angles - see Invariant Product: Sine Law Disguised.
7) Interestingly, Burton (1949) in deriving the Law of Cosines from the Sine Rule, during his derivation, arrives at the same Cosine-Sine Rule given above - see Using the law of sines.
(So as mentioned at the top of this page, this curious result is not new or original.)
Reference
Burton, L. J. (1949). The Laws of Sines and Cosines. The American Mathematical Monthly. 56 (8): 550–551.
Some Related Links
Invariant Product related to Triangles on the sides of a Triangle
Triangle Area Formula in terms of angles, r & R
Area Formula for Quadrilateral in terms of its Diagonals
Square Trigonometry
The Gielis Super Shape Formula
A Forgotten Similarity Theorem?
Pythagorean Quadrilaterals & Some Properties
A Generalization of the Theorem of Pythagoras by Hannes du Plooy
Carnot-Bottema Perpendicularity Theorem & Some Generalizations
Thabit's Generalization of the Theorem of Pythagoras
The theorem of Hippocrates (470 – c. 410 BC)
Cross's (Vecten's) theorem & generalizations to quadrilaterals
Intersecting Circles Investigation (with proof of Pythagoras)
Cyclic Kepler Quadrilateral Conjectures
Golden Quadrilaterals: Generalizing the concept of a golden rectangle
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices
Three Overlapping Circles (Haruki's Theorem)
A Generalization of the British Flag theorem to Semi-Regular Angle-2n-gons
British Mathematical Olympiad problem: Conic Generalization
Some External Links
Trigonometry (Wikipedia)
Law of cosines (Wikipedia)
Ceva's theorem (Wikipedia)
Trigonometric Form of Ceva's Theorem (Cut The Knot)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers as well as books on problem solving can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Michael de Villiers, Created with JavaSketchpad, July 2010; Updated to WebSketchpad, 13 May 2025; 31 July 2025; 23 June 2026.