The Law of Sines

If we label our triangle in the usual way, then we can give 3 different formulas for the area, depending on which two sides and the included angle are used:

Since these three formulas give the same area, the expressions on the right are all equal, so each of the following must be true:

These simplify to:

They can be rewritten as ratios:

or more compactly as the Law of Sines:

The following examples illustrate the use of this Law of Sines for finding missing parts of triangles.

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