Section 9-3 The Law of Sines

Recall…  When there are several methods for solving a problem, a comparison of the solutions can lead to new and useful results. From Section 9-2, we know three ways to find the area K of ∆ABC, depending on which pair of sides is known.

The Law of Sines  Area of a triangle: K = ½bc sin A = ½ac sin B = ½ab sin C  If each of these expressions is divided by ½abc, we obtain the law of sines.  The Law of Sines  In ∆ABC,

The Law of Sines  If we know two angles and a side of a triangle, then we can use the law of sines to find the other sides.

Activity 1 p. 346  Draw A with measure 30°. Along one side of A, locate point C 10 cm from point A. For each of the following compass settings, draw a large arc. Then tell whether the arc crosses the other ray of A and, if so, in how many points.

Activity 1  This activity demonstrates that when you are given the lengths of two sides of a triangle and the measure of the nonincluded angle (SSA), it may be possible to construct no triangle, one triangle, or two triangles. For this reason, the SSA situation is called the ambiguous case.

Activity 2 p. 346 a. If A = 30°, b = 10, and a = 4, find B b. If A = 30°, b = 10, and a = 5, find B c. If A = 30°, b = 10, and a = 6, find B b = 10 30° A B C

p. 347 problem 1 a. a < b sin A b. a = b sin A c. b sin A < a < b d. a > b

Example  Use the law of sines to determine whether there are 0, 1, or 2 triangles possible. a. a = 2, b = 4, A = 22° b. b = 3, c = 6, B = 30° c. a = 7, c = 5, A = 68° d. b = 4, c = 3, C = 76°

Example 1. Solve ∆RST if S = 40°, r = 30, and s = 20. Give angle measures to the nearest tenth of a degree and lengths to three significant digits.

Example 2. In ∆ABC, cos A = ½, cos B = -¼, and a = 6. Find the value of b in simplest radical form.