What is the sine rule?
The sine rule states that in any triangle, each side divided by the sine of its opposite angle gives the same value: a/sin A = b/sin B = c/sin C. It works for all triangles, not just right-angled ones.
// maths › Non-Right-Angle Trigonometry
Sine rule
Use the sine rule a over sin A equals b over sin B to find a missing side or angle in any triangle when you have a matching side and opposite angle, in degrees or radians, with a triangle diagram showing the equal ratios.
a / sin A = b / sin B = c / sin C
Angle unit
Frequently asked questions
When should I use the sine rule?
Use it when you know a side and its opposite angle, plus one more side or angle — typically two angles and a side (AAS/ASA) or two sides and a non-included angle (SSA).
Can you give a worked example?
With a = 7 opposite A = 30°, to find b opposite B = 45°: b = a·sin B / sin A = 7·sin 45° / sin 30° ≈ 9.90. The same angles in radians are 30° ≈ 0.5236 rad and 45° ≈ 0.7854 rad; the Degrees/Radians toggle switches the inputs and the working.
What is the ambiguous case?
When finding an angle from two sides and a non-included angle (SSA), the inverse sine can give two valid answers — an acute angle and its obtuse supplement — so up to two triangles may fit.
Where is this used in real life?
Surveyors and navigators use it to find distances they cannot measure directly, engineers use it in trusses and structures, and it appears in astronomy and triangulation for positioning.