# 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.

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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.

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

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.

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

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.

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

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.

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

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.

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.

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

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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