Triangle identities · proof ·what it is and when to use it

Proof: a/sin A = b/sin B

Law of sines — proved by one altitude, two right triangles. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

The sine rule is the right-triangle definition of sine applied twice to the same height, so nothing about a general triangle is assumed anywhere.

1. h = b·sin(A)
   In the right triangle at A, the perpendicular from C is opposite angle A.
2. h = a·sin(B)
   The same perpendicular is opposite angle B in the right triangle at B.
3. b·sin(A) = a·sin(B)
   Both expressions are the same height h.
4. a/sin(A) = b/sin(B)
   Divide both sides by sin(A)·sin(B). This is the checked claim. · Law of sines
5. a/sin(A) = b/sin(B) = c/sin(C)
   The same argument with the altitude from A instead of C gives B ↔ C, so the third ratio joins the chain.

The sampler checks the two-ratio claim on real triangles (A + B + C = 180°, sides measured from the corners); the last line is an argument, not a second measurement.

Where the proof stops applying

The same claim, checked numerically

A proof is not the same thing as a check, and this page shows both: the reasoning above, and the first few triangles fed to the same engine behind the verifier. If they ever disagreed, the data would be broken — the build would fail before publishing (accuracy policy).

Related

• How would I find a/sin A = b/sin B myself? — the derivation, which is a different question from the proof.
• Law of sines: when to use it — the practical side.
• All triangle identities · proof index

sampled at 800 real trianglestriangles only: A + B + C = 180°Source: OpenStax Precalculus, Ch. 7.4 (Law of Sines / Law of Cosines) · Paul's Online Math Notes — trig cheat sheet · revised 2026-09-28 ·how we check ·accuracy policy ·report an error