8.2.2 Law of Sines and Law of Cosines Key Objectives • Use the Law of Sines and the Law of Cosines to solve triangles. Theorems, Postulates, Corollaries, and Properties • The Law of Sines For any △ ABC with side lengths a , b , and c ,
= c sin sin sin = A a B b
C
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• The Law of Cosines For any △ ABC with side lengths a , b , and c , a 2 = b 2 + c 2 − 2 bc cos A ; b 2 = a 2 + c 2 − 2 ac cos B ; and c 2 = a 2 + b 2 − 2 ab cos C . Example 1 Finding Trigonometric Ratios for Acute and Obtuse Angles The trigonometric functions of both acute and obtuse angles are obtained using a calculator in this example.
To obtain the sine of 127 ° , an obtuse angle, enter 127 on the keypad and then push the “sine” key. The result rounds to 0.80. The tangent of 54 ° , an acute angle, is found to be 1.38, rounded to the nearest hundredth. The cosine of 101 ° , an obtuse angle, rounds to a value of − 0.19. Some sines, cosines, and tangents have negative values. The Law of Sines applies to any triangle, not just right triangles. For any △ ABC with side lengths a , b , and c , = = A a B b C c sin sin sin , where a is the length of the side opposite ∠ A , etc. The length of a side in a triangle is obtained in this example using the Law of Sines. The Law of Sines is used to set up the proportion sin Q is to the length of the known side, PR , as sin R is to the length of the unknown side, PQ . Remember that the ratio is of the sine of an angle and the length of the side opposite it. Substitute the known values, and r for the length of the unknown side, PQ . Then cross multiply and solve for r . The solution for r gives a value of approximately 4.7 units for the length of PQ .
Example 2 Using the Law of Sines
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