EXAMPLE 1 Solve a triangle for the SAS

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EXAMPLE 1 Solve a triangle for the SAS case ABC with a = 11,

EXAMPLE 1 Solve a triangle for the SAS case 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 – 2 ac cos B Law of cosines b 2 = 112 + 142 – 2(11)(14) cos 34° Substitute for a, c, and B. b 2 61. 7 Simplify. b 2 61. 7 7. 85 Take positive square root.

EXAMPLE 1 Solve a triangle for the SAS case Use the law of sines

EXAMPLE 1 Solve a triangle for the SAS case Use the law of sines to find the measure of angle A. sin A sin B Law of sines = a b sin 34° sin A = 7. 85 11 11 sin 34° sin A = 7. 85 A sin – 1 0. 7836 Substitute for a, b, and B. 0. 7836 Multiply each side by 11 and Simplify. 51. 6° Use inverse sine. The third angle C of the triangle is C = 94. 4°. ANSWER In ABC, b 7. 85, A 180° – 34° – 51. 6° 51. 68, and C 94. 48.

EXAMPLE 2 Solve a triangle for the SSS case ABC with a = 12,

EXAMPLE 2 Solve a triangle for the SSS case 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 – 2 ac cos B Law of cosines 272 = 122 + 202 – 2(12)(20) cos B Substitute. 272 = 122 + 202 = cos B – 2(12)(20) – 0. 3854 cos B B cos – 1 (– 0. 3854) Solve for cos B. Simplify. 112. 7° Use inverse cosine.

EXAMPLE 2 Solve a triangle for the SSS case Now use the law of

EXAMPLE 2 Solve a triangle for the SSS case Now use the law of sines to find A. sin A sin B = a b Law of sines sin A sin 112. 7° = 12 27 Substitute for a, b, and B. 12 sin 112. 7° sin A = 27 A sin– 1 0. 4100 24. 2° Multiply each side by 12 and simplify. Use inverse sine. The third angle C of the triangle is C = 43. 1°. ANSWER In ABC, A 24. 2, B 180° – 24. 2° – 112. 7° 112. 7, and C 43. 1.

EXAMPLE 3 Use the law of cosines in real life Science Scientists can use

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 Law of cosines

EXAMPLE 3 Use the law of cosines in real life SOLUTION Law of cosines b 2 = a 2 + c 2 – 2 ac cos B 3162 = 1552 + 1972 – 2(155)(197) cos B Substitute. 3162 = 1552 + 1972 = cos B – 2(155)(197) B Solve for cos B. – 0. 6062 cos B Simplify. cos – 1 (– 0. 6062) 127. 3° Use inverse cosine. ANSWER The step angle B is about 127. 3°.

for Examples 1, 2, and 3 GUIDED PRACTICE Find the area of ABC. 1.

for Examples 1, 2, and 3 GUIDED PRACTICE 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 – 2 ac cos B Law of cosines b 2 = 82 + 102 – 2(8)(10) cos 48° Substitute for a, c, and B. b 2 57 Simplify. b 2 57 7. 55 Take positive square root.

for Examples 1, 2, and 3 GUIDED PRACTICE Use the law of sines to

for Examples 1, 2, and 3 GUIDED PRACTICE Use the law of sines to find the measure of angle A. sin A sin B Law of sines = a b sin 48° sin A = 7. 55 8 8 sin 48° sin A = 7. 55 A sin – 1 0. 7836 Substitute for a, b, and B. 0. 7874 Multiply each side by 8 and simplify. 51. 6° Use inverse sine. The third angle C of the triangle is C = 79. 8°. ANSWER In ABC, b 7. 55, A 180° – 48° – 52. 2°, and C 94. 8°.

GUIDED PRACTICE for Examples 1, 2, and 3 Find the area of ABC. 2.

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 – 2 ac cos B Law of cosines 162 = 142 + 92 – 2(14)(9) cos B Substitute. 162 = 142 + 92 – 2(14)(9) Solve for cos B. = cos B

for Examples 1, 2, and 3 GUIDED PRACTICE – 0. 0834 cos B B

for Examples 1, 2, and 3 GUIDED PRACTICE – 0. 0834 cos B B cos – 1 (– 0. 0834) Simplify. 85. 7° Use inverse cosine. Use the law of sines to find the measure of angle A. sin A = sin B Law of sines a b sin A sin 85. 2° = 14 16 14 sin 85. 2° sin A = 16 Substitute for a, b, and B. 0. 8719 Multiply each side by 14 and simplify.

for Examples 1, 2, and 3 GUIDED PRACTICE A sin– 1 0. 8719 60.

for Examples 1, 2, and 3 GUIDED PRACTICE A sin– 1 0. 8719 60. 7° Use inverse sine. The third angle C of the triangle is C 34. 1°. 180° – 85. 2° – 60. 7° = ANSWER In ABC, A 60. 7°, B 85. 2°, and C 34. 1°.

GUIDED PRACTICE for Examples 1, 2, and 3 3. What If? In Example 3,

GUIDED PRACTICE for Examples 1, 2, and 3 3. What If? In Example 3, suppose that a = 193 cm, b = 335 cm, and c = 186 cm. Find the step angle θ. SOLUTION Law of cosines b 2 = a 2 + c 2 – 2 ac cos B 3352 = 1932 + 1862 – 2(193)(186) cos B Substitute. 3352 = 1932 + 1862 = cos B – 2(193)(186) – 0. 5592 B cos – 1 (– 0. 5592) ANSWER cos B 127° Solve for cos B. Simplify. Use inverse cosine. The step angle B is about 124°.