## Cosine-Sine Angle Rule

Around 1998, I conjectured the following trigonometric result on the basis of the side-angle duality that often appears in Euclidean geometry. It was specifically inspired by the (not so well-known, except in problem-solving circles) trigonometric version of Ceva's theorem. The validity of this conjecture was quickly confirmed by construction and dragging with the dynamic geometry software Sketchpad, as well as by a one line proof. Though this curious-looking result is in all likelihood not new or original, it doesn't seem to be well known or appear in standard mathematical textbooks.

Result: In any triangle ABC, (sin A)2 = (sin B)2 + (sin C)2 - 2 sin B sin C cos A.

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } Cosine-Sine Angle Rule

Explanation (proof):
Can you explain why (prove) this result true? Click on the provided HINT in the sketch if you get stuck.

Created online by Michael de Villiers, around 2005, updated 25 May 2017.