Triangle identities · derivation ·proof
Where ½ a b sin C = ½ b c sin A comes from
Divide to get the sine rule. This identity is how the sine rule falls out: cancel the common factor in the area equations.
½ a b sin C = ½ b c sin A
Building it step by step
- ½ a b sin C = ½ b c sin AEqual areas. · Triangle area in two pairings
- divide by ½ bCommon factor.
- a sin C = c sin A → a/sin A = c/sin CRearrange — the sine rule for two of the three pairs.
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- ½ a b sin C = ½ b c sin A — Triangle area in two pairings
That list is the argument for learning fewer formulas: most of this catalogue is a short chain away from the unit circle and the sum formulas.
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from ½ a b sin C = ½ b c sin A and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the triangle area in two pairings page.