What is the law of sines
The law of sines states that each side of a triangle divided by the sine of its opposite angle is constant: a ÷ sin A = b ÷ sin B. Rearranged to solve for an unknown side, a = b · sin A ÷ sin B. For b = 10, A = 30°, B = 90°, a = 10 × 0.5 ÷ 1 = 5. Recovering the angle from its sine is the job of the arcsine calculator, and it is where the ambiguous case bites.
How to use this calculator
Enter the known side b, the angle A opposite the side you want, and the angle B opposite the known side. The tool returns side a. Angles are in degrees.
Pairing each angle with the side opposite it is the step people most often get wrong. If your angles are in radians, convert them to degrees first.
When to use it
The law of sines is ideal when you know an angle-side pair plus one more angle. When you instead know two sides and the included angle, reach for the law of cosines. Both rely on the sine function.
Since a triangle's angles always sum to 180°, knowing two angles gives you the third for free, which means one side is enough to unlock the whole triangle.
Frequently asked questions
- What is the law of sines formula?
- a ÷ sin A = b ÷ sin B = c ÷ sin C. To find a side: a = b · sin A ÷ sin B.
- What is side a when b = 10, A = 30°, B = 90°?
- a = 5, from a = 10 × sin 30° ÷ sin 90° = 10 × 0.5 ÷ 1.
- When does the law of sines fail?
- It needs sin B ≠ 0. An angle B of 0° or 180° makes sin B = 0, so no side can be computed.
- What is the ambiguous case?
- When you know two sides and a non-included angle (SSA), the law of sines can allow two different triangles, one acute and one obtuse. Check whether the second solution's angles still sum to 180° before accepting it.
- How is it different from the law of cosines?
- The law of sines uses an angle-side pair plus another angle; the law of cosines uses two sides and the included angle.