Enter two triangle sides and one opposite angle to solve unknown sides and angles
Given: Side a, Side b and Angle A (Solve for Side c)
Side a
Side b
Angle A (°)
Angle B (°)
Calculation Result
Step-by-Step Derivation
Law of Sines Formula
a / sin A = b / sin B = c / sin C = 2R
Where a, b, c are triangle sides, A, B, C are opposite angles, R is the circumcircle radius. The Law of Sines describes the fixed proportion between each side and the sine value of its opposite angle.
⚠Notice: SSA two sides and one opposite angle may produce 0, 1 or 2 ambiguous solutions. Input angles in degrees; calculation uses radian internally.
What Is the Law of Sines?
The Law of Sines defines the proportional relationship between triangle side lengths and the sine values of their opposite angles. It is an essential tool for solving triangles, widely used in surveying, navigation and physics calculations.
SSA Ambiguous Case
Given sides a, b and angle A a opposite angle, if a < b and a > b·sin(A), two valid triangles exist one acute, one obtuse solution.
Solution Steps
① Solve angle: sin B = b·sin A / a; ② Calculate angle C = 180° - A - B; ③ Solve side c = a·sin C / sin A.
Law of Cosines Complement
Use Law of Sines for two sides one angle; use Law of Cosines for two sides included angle or three known sides. Combined they solve any triangle.
Circumcircle Radius
From a/sin A = 2R, R equals triangle circumscribed circle radius. Formula: R = a/(2 sin A) for verification and geometry calculation.
💡 Example Tutorial: Given a=5, b=7, A=30°. sin B = 7×sin 30°/5 = 7×0.5/5 = 0.7. B₁ = arcsin(0.7) ≈ 44.4° or B₂ = 180°-44.4° = 135.6°. A+B₂=165.6°<180°, both solutions valid.
Application Scenarios
Law of SinesTriangle SolverSurveying MathExam ReviewGeometry Competition
Frequently Asked Questions
What is the Law of Sines formula?▼
Law of Sines formula: a/sin A = b/sin B = c/sin C = 2R, where a, b, c are triangle sides, A, B, C are corresponding opposite angles, and R is the circumscribed circle radius. It defines the proportional relationship between triangle sides and the sine of their opposite angles.
How many solutions does SSA ambiguous case have?▼
SSA Side-Side-Angle may have 0, 1 or 2 solutions known as ambiguous case. No solution if opposite side < adjacent side × sin(known angle); one right angle solution if equal; one solution if opposite side ≥ adjacent side; two valid solutions if opposite side is between adjacent side × sin(angle) and adjacent side.
When to use the Law of Sines?▼
The Law of Sines applies to SSA two sides and one opposite angle, AAS and ASA two angles and one side. It works complementary with the Law of Cosines for full triangle solving.
Difference between Law of Sines and Law of Cosines?▼
Law of Sines: a/sin A = b/sin B = c/sin C, used for two sides one angle or two angles one side. Law of Cosines: c² = a² + b² - 2ab cos C, used for two sides included angle or three known sides. Both complete triangle solving together.
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