Triangle identities · trigonometric identity
a/sin A = b/sin B — Law of sines
When to use it
You have one complete pair — a side together with the angle opposite it — and you need an angle or a side from a second pair. It is also the formula that produces the ambiguous case: with two sides and a non-included angle, sin B can come from two different angles.
Why it is true
Drop the perpendicular from C onto AB. In the two right triangles that height is h = b·sin A and also h = a·sin B, so b·sin A = a·sin B, and dividing both sides by sin A·sin B leaves a/sin A = b/sin B.
The full line-by-line version is on the proof page for law of sines; the “how would I find this myself” version is in the derivation.
Where it comes from
same area, two different base/height pairings. Start from the area formula this site already publishes and cancel what is common — the ratio falls out without drawing a height at all.
- ½·a·b·sin(C) = ½·b·c·sin(A)Both sides are the area of the same triangle. · Triangle area in two pairings
- a·sin(C) = c·sin(A)Divide both sides by ½b.
- a/sin(A) = c/sin(C)Divide by sin(A)·sin(C).
Worked examples
- A = 30°, a = 6, b = 10 — solve the triangleTwo triangles: B ≈ 56.44° with C ≈ 93.56° and c ≈ 11.98, or B ≈ 123.56° with C ≈ 26.44° and c ≈ 5.34.
Checked on these triangles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| Triangle (A) | Left side | Right side | Agree |
|---|---|---|---|
| 7.5° | 12.419529 | 12.419529 | yes |
| 15° | 3.006778 | 3.006778 | yes |
| 18° | 3.731834 | 3.731834 | yes |
| 22.5° | 5.219878 | 5.219878 | yes |
| 30° | 6.408182 | 6.408182 | yes |
| 37° | 4.162155 | 4.162155 | yes |
| 45° | 3.220779 | 3.220779 | yes |
| 53° | 3.815035 | 3.815035 | yes |
| 60° | 4.579938 | 4.579938 | yes |
| 67.5° | 3.918527 | 3.918527 | yes |
| 75° | 3.75013 | 3.75013 | yes |
| 90° | 4.816443 | 4.816443 | yes |
| 120° | 6.394203 | 6.394203 | yes |
| 135° | 7.774097 | 7.774097 | yes |
| 150° | 12.021655 | 12.021655 | yes |
The letters are not free
a, b, c and A, B, C here are one triangle's sides and angles: A + B + C = 180° and the sides follow the sine rule. Feed the same six letters with random values and the equation fails — that is a property of the formula, not a bug in the tool.
The mistake students make
Pairing a side with the angle next to it instead of the angle opposite it. In a/sin A the letter A must name the angle across from side a; with the wrong pairing the two ratios are simply not equal.
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for triangle identities.
Related identities
- a² = b² + c² − 2bc·cos ALaw of cosinesdetailsproof
- ½ a b sin C = ½ b c sin ATriangle area in two pairingsdetailsproof
- A + B + C = 180°Angles of a triangle add to 180°detailsproof
Category hub: Triangle identities · all identities: /identities