What is the law of sines?

The law of sines states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C).

What is the ambiguous case?

When given two sides and a non-included angle (SSA), there may be zero, one, or two possible triangles. This is the ambiguous case of the law of sines.

When should I use law of sines vs law of cosines?

Use law of sines for AAS, ASA, or SSA problems. Use law of cosines for SAS or SSS problems.

Law of Sines Calculator

Use the law of sines to find a missing side. Given side a, angle A, and angle B, find side b.

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Side a

Angle A (degrees)

deg

Angle B (degrees)

deg

Side B

17.3205

Angle C90.00 deg

Side C20.0000

View formula & how it works →

Law of Sines

Formula

a / sin(A) = b / sin(B) = c / sin(C)

This law relates the sides of any triangle to the sines of their opposite angles.

Solving for Side b

b = a × sin(B) / sin(A)

When to Use

Use the Law of Sines when you know:

Two angles and one side (AAS or ASA)

Two sides and an angle opposite one of them (SSA, but watch for the ambiguous case)

Example Calculation

Find side b: a = 10, angle A = 30°, angle B = 60°.

01b = 10 × sin(60°) / sin(30°)
02= 10 × 0.8660 / 0.5
03= 17.3205
04Angle C = 180 - 30 - 60 = 90°

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