Technical Mathematics, Sixth Edition

8.3. Law of Sines

We cannot use the trigonometric functions directly to solve an oblique triangle, but we will use them to derive the law of sines, which can be used for any triangle.

8.3.1. Derivation

We will derive the law of sines for an oblique triangle in which all three angles are acute such as in Fig. 8-18. We start by breaking the given triangle into two right triangles by drawing altitude h to side AB.

Then right triangle ACD,

And right triangle BCD,

So

b sin A = a sin B

Dividing by sin A sin B, we have

Figure 8.18. Derivation of the law of sines.

Similarly, drawing altitude j to side AC, and using triangles BEC and AEB, we get or

j = a sin C = c sin A

or

Combining this with the previous result, we obtain the following equation:

NOTE

The sides of a triangle are proportional to the sines of the opposite angles.

We have just derived the law of sines for a triangle having all acute angles. The law of sines also holds when one of the angles is obtuse, and the derivation is ...