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EXAMPLE 1 Solve a triangle for the SAS case Solve ABC with a = 11, c = 14, and B = 34°. SOLUTION Use the law of cosines to find side length b. b 2 = a 2 + c 2 – 2ac cos B b 2 = 11 2 + 14 2 – 2(11)(14) cos 34° b 2 61.7 7.85 Law of cosines Substitute for a, c, and B. Simplify. Take positive square root.

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EXAMPLE 1 Solve a triangle for the SAS case Use the law of sines to find the measure of angle A. sin A a sin B b = sin A 11 = sin 34° 7.85 sin A = 11 sin 34° 7.85 0.7836 A sin –1 0.7836 51.6° Law of sines Substitute for a, b, and B. Multiply each side by 11 and Simplify. Use inverse sine. The third angle C of the triangle is C 180° – 34° – 51.6° = 94.4°. In ABC, b 7.85, A 51.68, and C 94.48. ANSWER

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EXAMPLE 2 Solve a triangle for the SSS case Solve ABC with a = 12, b = 27, and c = 20. SOLUTION First find the angle opposite the longest side, AC. Use the law of cosines to solve for B. b 2 = a 2 + c 2 – 2ac cos B 27 2 = 12 2 + 20 2 – 2(12)(20) cos B 27 2 = 12 2 + 20 2 – 2(12)(20) = cos B – 0.3854 cos B B cos –1 (– 0.3854) 112.7° Law of cosines Substitute. Solve for cos B. Simplify. Use inverse cosine.

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EXAMPLE 2 Solve a triangle for the SSS case Now use the law of sines to find A. sin A a = sin B b sin A 12 sin 112.7° 27 = sin A= 12 sin 112.7° 27 0.4100 A sin –1 0.4100 24.2° Law of sines Substitute for a, b, and B. Multiply each side by 12 and simplify. Use inverse sine. The third angle C of the triangle is C 180° – 24.2° – 112.7° = 43.1°. In ABC, A 24.2, B 112.7, and C 43.1. ANSWER

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EXAMPLE 3 Use the law of cosines in real life Science Scientists can use a set of footprints to calculate an organism’s step angle, which is a measure of walking efficiency. The closer the step angle is to 180°, the more efficiently the organism walked. The diagram at the right shows a set of footprints for a dinosaur. Find the step angle B.

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EXAMPLE 3 Use the law of cosines in real life SOLUTION b 2 = a 2 + c 2 – 2ac cos B 316 2 = 155 2 + 197 2 – 2(155)(197) cos B 316 2 = 155 2 + 197 2 – 2(155)(197) = cos B – 0.6062 cos B B cos –1 (– 0.6062) 127.3° Use inverse cosine. Simplify. Solve for cos B. Substitute. Law of cosines The step angle B is about 127.3°. ANSWER

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GUIDED PRACTICE for Examples 1, 2, and 3 Find the area of ABC. 1. a = 8, c = 10, B = 48° SOLUTION Use the law of cosines to find side length b. b 2 = a 2 + c 2 – 2ac cos B b 2 = 8 2 + 10 2 – 2(8)(10) cos 48° b 2 57 7.55 Law of cosines Substitute for a, c, and B. Simplify. Take positive square root.

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GUIDED PRACTICE for Examples 1, 2, and 3 Use the law of sines to find the measure of angle A. sin A a sin B b = sin A 8 = sin 48° 7.55 sin A = 8 sin 48° 7.55 0.7874 A sin –1 0.7836 51.6° Law of sines Substitute for a, b, and B. Multiply each side by 8 and simplify. Use inverse sine. The third angle C of the triangle is C 180° – 48° – 52.2° = 79.8°. In ABC, b 7.55, A 52.2°, and C 94.8 °. ANSWER

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16 2 = 14 2 + 9 2 – 2(14)(9) cos B GUIDED PRACTICE for Examples 1, 2, and 3 Find the area of ABC. 2. a = 14, b = 16, c = 9 SOLUTION First find the angle opposite the longest side, AC. Use the law of cosines to solve for B. b 2 = a 2 + c 2 – 2ac cos B 16 2 = 14 2 + 9 2 – 2(14)(9) = cos B Law of cosines Substitute. Solve for cos B.

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GUIDED PRACTICE for Examples 1, 2, and 3 – 0.0834 cos B B cos –1 (– 0.0834) 85.7° Simplify. Use inverse cosine. sin A a = sin B b sin A 14 sin 85.2° 16 = sin A= 14sin 85.2° 16 0.8719 Law of sines Substitute for a, b, and B. Multiply each side by 14 and simplify. Use the law of sines to find the measure of angle A.

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GUIDED PRACTICE for Examples 1, 2, and 3 The third angle C of the triangle is C 180° – 85.2° – 60.7° = 34.1°. A sin –1 0.8719 60.7° Use inverse sine. In ABC, A 60.7°, B 85.2°, and C 34.1°. ANSWER

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GUIDED PRACTICE for Examples 1, 2, and 3 SOLUTION b 2 = a 2 + c 2 – 2ac cos B 335 2 = 193 2 + 186 2 – 2(193)(186) cos B 335 2 = 193 2 + 186 2 – 2(193)(186) = cos B – 0.5592 cos B B cos –1 (– 0.5592) 127° Use inverse cosine. Simplify. Solve for cos B. Substitute. Law of cosines 3. What If? In Example 3, suppose that a = 193 cm, b = 335 cm, and c = 186 cm. Find the step angle θ. The step angle B is about 124°. ANSWER

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Essential Question: What is the law of cosines, and when do we use it?

Essential Question: What is the law of cosines, and when do we use it?

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