Geometry & measures · Trigonometry (sine & cosine rules)
Chapter 1 · 3
The idea
The cosine rule
Why a² = b² + c² − 2bc cos A is Pythagoras with a correction term, when to choose it over the sine rule, and the rearrangement that finds an angle from three sides.
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Geometry & measures · Trigonometry (sine & cosine rules)
The cosine rule
Why a² = b² + c² − 2bc cos A is Pythagoras with a correction term, when to choose it over the sine rule, and the rearrangement that finds an angle from three sides.
Why it works
Pythagoras with a correction term
It is Pythagoras with a correction term. For a triangle with sides , , and the angle opposite :When , and the correction vanishes, leaving — ordinary Pythagoras. When is acute, , so we subtract something and comes out shorter than Pythagoras would predict; when is obtuse, is negative, the correction adds, and comes out longer. That built-in sign behaviour is why the rule handles obtuse triangles without any special case — unlike the sine rule.
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