Suppose is a triangle with sides of lengths , , and that are opposite their respective vertices , , and . Let , , and be the angles at , , and . The Law of Sines says that

.

When the appropriate information is known, the Law of Sines can be used to solve triangles.

Examples

, , and feet. (This is an AAS case.) Determine the remaining sides and angles.

A pole tilts toward the sun at an angle away from the vertical, and it casts a 22-ft shadow. The angle of elevation from the tip of the shadow to the top of the pole is . How long is the pole? (This is an ASA case.)

The Ambiguous Case (SSA or ASS)

A triangle is not necessarily uniquely determined in the SSA (ASS) case. There are three possibilities:

No solution

One solution

Two solutions

When solving triangles, always keep in mind that the longer sides lie opposite the larger angles.

Examples

inches, inches, and

inches, inches, and

meters, meters, and

Area Formulas

It is pretty easy to derive these formulas for the area of :

Area .

Examples

Find the area of a triangular lot with two sides of lengths 90 meters and 52 meters and an included angle of .

On a small lake, you swim from point A to point B at a bearing of N E, then to point C at a bearing of N W, and finally due south 800 meters to return to point A$. How far did you swim?