The formula
a ÷ sin A = b ÷ sin B = c ÷ sin C
Where it comes from
The law of sines says the ratio of a side to the sine of the angle facing it is constant across a triangle. That constant is not arbitrary: it equals the diameter of the circle through all three vertices, which is why the law works for any triangle at all.
How to work it out by hand
- You need a side with its opposite angle to start
- Work out the constant: that side divided by the sine of its angle
- Multiply the constant by the sine of any angle to get its opposite side
- Divide any side by the constant and take the arcsine for its angle
What is worth knowing
The ambiguous case is the trap, and it is worth understanding rather than memorising. When you know two sides and an angle that is not between them, the arcsine gives you one angle, but its supplement has the same sine — and sometimes both produce a valid triangle. Physically, the side you are swinging can reach the base line in two places. The law of cosines does not have this problem, which is a reason to prefer it when you have the choice. When you know two angles, there is no ambiguity at all, since the third angle is forced.