Section 1.5 Law of Sines

Objectives: 1. To prove the law of sines. 2. To identify the ambiguous case. 3. To solve triangles using the law of sines.

Law of Sines In ABC where a is the side length opposite A, b is opposite B, and c is opposite C, the following proportion exists: sin C c sin B b sin A a =

To apply the law of sines, you must know either the measures of two angles and a side (ASA or SAA) or the measures of two sides and angle opposite one of the given sides (SSA).

Example: Solve triangle ABC mA = 52, mB = 49, c = 16 mC = 180° - mA - mB mC = 180 - 52 - 49 mC = 79

Example: Solve triangle ABC mA = 52, mB = 49, c = 16 ° = 79 sin 16 52 a a(sin 79) = (16)(sin 52) (sin 79) (16)(sin 52) a = a ≈ 12.8

Example: Solve triangle ABC mA = 52, mB = 49, c = 16 ° = 79 sin 16 49 b b(sin 79) = (16)(sin 49) (sin 79) (16)(sin 49) b = b ≈ 12.3

Example: Solve triangle ABC mA = 52 mB = 49 mC = 79

SSA is called the ambiguous case because it does not always result in a unique triangle. The three possible cases follow. 1. There may be not be a solution. 2. There may be two solutions. 3. There may be a unique solution.

1. SSA – no solution No triangle exists when b  h. sin B = h/a a sin B = h If b  a sin B, there is no solution to SSA; no triangle is possible.

C a b h B b  a sin B

2. SSA – two solutions two triangles exist when h  b  a a sin B  b  a If a sin B  b  a, the solution to SSA can be either of two triangles.

C a b b h B a sin B  b  a

3. SSA – one solution one triangle exists when b = h b = a sin B or, one triangle exists when b  a. If b = a sin B or b  a, then the solution to SSA is a unique triangle.

C a b = h B b = a sin B

C a b h B b  a

Possible Cases for SSA: 1. b < a sin B - no solution 2. b = a sin B - 1 solution (rt. triangle) 3. b > a sin B and a > b - 2 solutions 4. b > a sin B and b > a - 1 solution

Practice: Give the number of triangles that satisfy the given information. c = 8, B = 35, C = 70 One

Practice: Give the number of triangles that satisfy the given information. a = 12, b = 9, B = 55 None

Practice: Give the number of triangles that satisfy the given information. b = 13, c = 14, B = 64 Two

Practice: Give the number of triangles that satisfy the given information. a = 10, c = 12, C = 73 One

Practice: Solve triangle ABC a = 6, c = 8, mC = 100 = sin 100 8 sin A 6 6(sin 100) = 8(sin A) 8 (6)(sin 100) sin A = sin A ≈ 0.7386

Practice: Solve triangle ABC a = 6, c = 8, mC = 100 mA ≈ sin-1 0.7386 mA ≈ 47.6 mB = 180 - mA - mC mB = 180 - 47.6 -100 mB ≈ 32.4

Practice: Solve triangle ABC a = 6, c = 8, mC = 100 = sin 100 8 sin 32.4 b b(sin 100°) = 8(sin 32.4°) sin 100° (8)(sin 32.4°) b = b ≈ 4.4

Practice: Solve triangle ABC mA ≈ 47.6 mB ≈ 32.4 mC = 100

Homework p. 27

►A. Exercises Specify the type of pattern. 1. a = 5, B = 48°, A = 68° 1. ASA 2. SAA 3. SSA

►A. Exercises Determine the number of triangles that satisfy the given information for ∆ABC. 1. a = 5, B = 48°, A = 68° 1. 0 2. 1 3. 2

►A. Exercises Determine the number of triangles that satisfy the given information for ∆ABC. 3. A = 12°, B = 106°, c = 10 1. 0 2. 1 3. 2

►A. Exercises Determine the number of triangles that satisfy the given information for ∆ABC. 5. a = 3, b = 7, C = 40° 1. 0 2. 1 3. 2

►A. Exercises Determine the number of triangles that satisfy the given information for ∆ABC. 7. b = 18, c = 9, B = 82° 1. 0 2. 1 3. 2

►A. Exercises Determine the number of triangles that satisfy the given information for ∆ABC. 9. a = 3, b = 7, c = 8 1. 0 2. 1 3. 2

►B. Exercises Solve the triangle. Round sides to the nearest tenth. Give angle measures to the nearest tenth of a degree. 11. A = 52°, B = 49°, c = 16 Find C.

►B. Exercises Solve the triangle. Round sides to the nearest tenth. Give angle measures to the nearest tenth of a degree. 13. B = 82°, b = 12, c = 6 Find C.

►B. Exercises Solve the triangle. Round sides to the nearest tenth. Give angle measures to the nearest tenth of a degree. 15. C = 15°, a = 14, c = 12 Find A.

►B. Exercises Solve the triangle. Round sides to the nearest tenth. Give angle measures to the nearest tenth of a degree. 17. B = 80°, a = 6, b = 2

■ Cumulative Review 25. If one acute angle of a right triangle has a measure of 0.2356 radians, what is the radian measure of the other acute angle?

■ Cumulative Review 26. Find two negative angles coterminal with 5/6.

■ Cumulative Review 27. In what quadrants is the cosecant negative, and why?

■ Cumulative Review In right triangle ABC, where C = 90°, suppose a = 3 and B = 40°. 28. Find b.

■ Cumulative Review In right triangle ABC, where C = 90°, suppose a = 3 and B = 40°. 29. Find cos A.