Law of sines and cosines

General triangle. In general form of the triangle  , the sides are denoted by , , and , and the angles are denoted by , , and .

Inscribed angle theorem. is called an inscribed angle, and is called a central angle. Then they have the following relationship:

Law of sines. Let is the radius of the circumscribed circle of , and let be the diameter of the circle.

Since by the inscribed angle theorem, we have ; thus, we obtain

Since the choice of is arbitrary, in general we have

Law of cosines. Let be perpendicular to . The square length  can be expressed in terms of the sides  and the angle  via the two different right triangles  and  .

 (2-1) (2-2)

Comparing the rightmost expressions in (2-1)-(2-2), we obtain . In general we have

Heron's formula. Let be the area of  , and let . By using via the law of cosines, we obtain

Department of Mathematics