Triangle identities · derivation ·proof
Where a = b·cos C + c·cos B comes from
combine the sine rule with the angle sum. A second route that does not draw an altitude: start from the ratio the sine rule gives and reduce the angle sum with the sine addition formula.
Building it step by step
- b·cos C + c·cos B = R(sin B cos C + sin C cos B)Replace b and c by the sine rule form, with R the common ratio a/sin A. · Law of sines
- = R·sin(B + C)Sine addition formula. · Sine of a sum
- = R·sin A = aB + C = π − A from the angle sum, and sin(π − A) = sin A. · Angles of a triangle add to 180°
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- a/sin A = b/sin B — Law of sines
- sin(A + B) = sin A cos B + cos A sin B — Sine of a sum
- A + B + C = 180° — Angles of a triangle add to 180°
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.
Where this route is used
A side has to be split into the two projections of the other sides — the step that turns a pair of sine-rule ratios into a single expression for a side, and the shortcut in proofs of the tangent rule. It is a relation between sides and angles, not a solver for the ambiguous case.
- simplify
- verify
Checked on these triangles
Deriving a formula and testing it are different things; this table is the test. The numbers come from the same engine as the verifier, computed when the site was built.
| Triangle (A) | Left side | Right side | Agree |
|---|---|---|---|
| 7.5° | 1.621074 | 1.621074 | yes |
| 15° | 0.778211 | 0.778211 | yes |
| 18° | 1.1532 | 1.1532 | yes |
| 22.5° | 1.997561 | 1.997561 | yes |
| 30° | 3.204091 | 3.204091 | yes |
| 37° | 2.504847 | 2.504847 | yes |
Neighbouring derivations
- a/sin A = b/sin B — Law of sines
- a² = b² + c² − 2bc·cos A — Law of cosines
- sin(A + B) = sin A cos B + cos A sin B — Sine of a sum
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from a/sin A = b/sin B and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the projection rule page.