773 8.1 The Law of Sines Solving a triangle with given information matching Case 1 or Case 2 requires using the law of sines, and solving a triangle with given information matching Case 3 or Case 4 requires using the law of cosines. Derivation of the Law of Sines To derive the law of sines, we start with an oblique triangle, such as the acute triangle in Figure 1(a) or the obtuse triangle in Figure 1(b). This discussion applies to both triangles. First, construct the perpendicular from B to side AC (or its extension). Let h be the length of this perpendicular. Then c is the hypotenuse of right triangle ADB, and a is the hypotenuse of right triangle BDC. In triangle ADB, sin A = h c , or h = c sin A. In triangle BDC, sin C = h a , or h = a sin C. Because h = c sin A and h = a sin C, we set these two expressions equal. a sin C = c sin A a sin A = c sin C Divide each side by sin A sin C. In a similar way, by constructing perpendicular lines from the other vertices, we can show that these two equations are also true. a sin A = b sin B and b sin B = c sin C This discussion proves the following theorem. Law of Sines In any triangle ABC, with sides a, b, and c, the following hold true. a sin A = b sin B , a sin A = c sin C , and b sin B = c sin C This can be written in compact form as follows. a sin A = b sin B = c sin C That is, according to the law of sines, the lengths of the sides in a triangle are proportional to the sines of the measures of the angles opposite them. In practice we can also use an alternative form of the law of sines. sin A a = sin B b = sin C c Alternative form of the law of sines NOTE When using the law of sines, a good strategy is to select a form that has the unknown variable in the numerator and where all other variables are known. This makes computation easier. We label oblique triangles as we did right triangles: side a opposite angle A, side b opposite angle B, and side c opposite angle C. A C B D h c a Acute triangle ABC (a) Figure 1 C B h c a A b D Obtuse triangle ABC (b)
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