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5 Triangle Trigonometry > 5.3 Law of Sines
5.3 Law of Sines
Introduction In Section 5.1 we saw how to solve right triangles. In this and the next section we consider two techniques for solving general triangles.
Law of Sines Consider the triangle ABC, shown in FIGURE 5.3.1, with angles a, ß, and ?, and corresponding opposite sides BC, AC, and AB. If we know the length of one side and two other parts of the triangle, we can then find the remaining three parts. One way of doing this is by the Law of Sines.
FIGURE 5.3.1 General triangle
THEOREM 5.3.1 The Law of Sines
Suppose angles a, ß, and ?, and opposite sides of length a, b, and c are as shown in FIGURE 5.3.1. Then
PROOF: Although the Law of Sines is valid for any triangle, we will prove it only for acute triangles—that is, a triangle in which all three angles a, ß, and ?, are less than 90°. As shown in FIGURE 5.3.2, let h be the length of the altitude from vertex A to side BC. Since the altitude is perpendicular to the base BC it determines two right triangles. Consequently, we can write
