The law of tangents is expressed in words as follows: In any triangle the difference between two sides is to their sum as the tangent of half the difference of the opposite angles is to the tangent of half their sum.

For any pair of sides—as side a and side b—the law may be expressed as follows:

For the triangle shown in figure 1-27, you know the lengths of two sides and the size of the angle between them. You can determine the sizes of the other two angles by applying the law of tangents as follows.

First note that you can determine the value of angles (B + C), because (B + C) obviously equals 180° – A, or 180° – 34°, or 146°. Now, if you know the sum of two values and the difference between the same two, you can determine each of the values as follows:

Now, you know the sum of (B + C). Therefore, if you could determine the difference, or (B – C), you could determine the sizes of B and C You can determine 12(B — C) from the law of tangents, written as follows:

One-half of (B + C) means one-half of 146°, or 73°. The tangent of 730 is 3.27085. The solution for 12(B – C) is therefore as follows:

(from table of natural tangents) 1/2 (B - C)

= 19°58’ (B – C) = 2(19058’) = 39°56’ Knowing both the sum (B + C) and the difference (B – C), you can now determine the sizes of B and C as follows: