3. Solving Triangles

There are three important congruence theorems from geometry that are usually abbreviated as ASA, SAS, and SSS. ASA, for example, tells us that if two angles and the included side of one triangle are equal to the respective two angles and included side of another triangle, the two triangles are congruent. Similar statements apply to the other two abbreviations. What these theorems tell us, is that if the right three parts of a triangle are known, the other three parts are fixed and they should be able to be found. Having partial information about a triangle and using it to find the rest of the information about the triangle is referred to as solving the triangle. The procedure for solving right triangles is demonstrated in Problem 24 of Exercises for Chapter 3 Section 1. This sections introduces two important laws of trigonometry that can be used for solving any triangle for which appropriate information is known.

Law of Sines

Consider the triangles in Figure 1a and Figure 1b. Since every triangle has at least two acute angles, we can assume we have picked one of them to call Since (see Problem 23, Exercises for Chapter 3 Section 1.) If is acute as in Figure 1a, then If is obtuse as in Figure 1b, then and In either case, Equating these values of h, we have or

Figure 1a

Figure 1b

Similarly, if is the angle at C and c is its opposite side, we can show that We are now prepared to state the Law of Sines: If a triangle has angles with opposite sides a, b, and c respectively, (See Figure 1c) then

Figure 1c

All of the examples and problems which follow are based on triangles labeled as in Figure 1c.

Example 1. If and c = 8, find angle C and sides a and b. (Given ASA.)

Solution: First, From Problems 6 and 19 of Exercises for Chapter III Section 1 we have and Now, from the Law of Sines, thus Similarly,

Figure 2

Law of Cosines Consider the complex numbers P and Q as shown in Figure 2. By Problem 33 of Exercises for Chapter II Sections 3, 4, and 5, we see that d, the distance between P and Q, is But Substituting these into the equation for d, squaring both sides, expanding, and collecting like terms, gives us

Using the Pythagorean Identity [See Problem 11, Exercise for Chapter III Section 1.] and the subtraction formula for the cosine [See Problem 17, Exercise for Chapter III Section 1.] this becomes

If is now relabeled as in Figure 3c, we get the three forms of the Law of Cosines:

depending on whether angle A, B or C, respectively is placed at the origin.

Example 2. If and b = 10, find side c and angles A, and B. (Given SAS.)

Solution: From Problem 5 of Exercises for Chapter 3 Section 1, Then, from the Law of Cosines,

Thus, From the Law of Sines Since