The Law of Sines

Every triangle, right angle or not, has a hidden consistency: bigger angles sit across from longer sides, and that relationship is exact. The Law of Sines pins it down as a/sin A = b/sin B = c/sin C, where lowercase letters are side lengths and capital letters are the angles directly across from them. It works because every triangle can be split into two right triangles by dropping an altitude from one vertex to the opposite side.

Drop that altitude from vertex C down to side c, and call its height h. Now you have two right triangles sharing h. In one, sin A = h/b, so h = b sin A. In the other, sin B = h/a, so h = a sin B. Both expressions equal h, so b sin A = a sin B, which rearranges into a/sin A = b/sin B. Repeat the drop from a different vertex and C joins the chain. The whole law is just this one trick done twice.

Use the Law of Sines whenever you know an angle and its opposite side, plus one more piece of information, either another angle or another side. It sits next to the Law of Cosines, which takes over when you only have sides, or an angle squeezed between two known sides with nothing opposite it. Together the two laws solve every triangle; this one is the lighter tool, worth reaching for first whenever an angle-side pair is sitting in front of you.