Definition: The law of sines 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.
When do you use it?
For triangles that do not have a right angle, when you know two angles and one side (AAS or ASA), or two sides and an angle opposite one of them (SSA). In each case you know one complete side-and-opposite-angle pair.
How do you apply it?
Set up a proportion with the known pair on one side and the unknown on the other.
Example. In triangle ABC, A = 48 degrees, B = 71 degrees, and a = 15. Then C = 61 degrees. To find b: b / sin 71 degrees = 15 / sin 48 degrees. So b = 15(0.9455) / 0.7431, about 19.1. To find c: c = 15(sin 61 degrees) / sin 48 degrees = 15(0.8746) / 0.7431, about 17.7.
What can go wrong?
With SSA, there may be two triangles, one, or none, known as the ambiguous case. If the sine ratio gives a value above 1, no triangle exists. Another mistake is to pair a side with the wrong angle: each side goes with the angle opposite it. Calculators must be in degree mode.