19. Law of Sines

Introduction In this section, we will solve (find all the sides and angles of) oblique triangles – triangles that have no right angles. As standard notation, the angles of a triangle are labeled A, B, and C, and their opposite sides are labeled a, b, and c. To solve an oblique triangle, we need to know the measure of at least one side and any two other measures of the triangle—either two sides, two angles, or one angle and one side.

4 cases 1. Two angles and any side are known (AAS or ASA) 2. Two sides and an angle opposite one of them are known (SSA) 3. Three sides are known (SSS) 4. Two sides and their included angle are known (SAS) The first two cases can be solved using the Law of Sines, the last two cases require the Law of Cosines.

CASE 1: ASA or SAA Law of sines S A A ASA S AA SAA

S S A CASE 2: SSA Law of sines

S S A CASE 3: SAS Law of cosines

S S S CASE 4: SSS Law of cosines

Law of Sines A B C a bc

Remember ……

Case 1 - AAS For the triangle below C = 102 , B = 29 , and b = 28 feet. Solve the triangle

Example AAS - Solution The third angle of the triangle is A = 180  – B – C = 180  – 29  – 102  = 49 . By the Law of Sines, you have.

Example AAS – Solution Using b = 28 produces and cont’d

Case 1 - ASA A B C c a b C=70 o b=44.1 a=32.7

Example A 46-foot telephone pole tilted at an angle of from the vertical casts a shadow on the ground. Find the length of the shadow to the nearest foot when the angle of elevation to the sun is Draw a diagram Draw Then find the

Example cont Cross products Use a calculator. Law of Sines Answer: The length of the shadow is about 75.9 feet. Divide each side by sin Since you know the measures of two angles of the triangle, and the length of a side opposite one of the angles you can use the Law of Sines to find the length of the shadow.

Example A 5-foot fishing pole is anchored to the edge of a dock. If the distance from the foot of the pole to the point where the fishing line meets the water is 45 feet, about how much fishing line that is cast out is above the surface of the water? Answer: About 42 feet of the fishing line that is cast out is above the surface of the water.

Area of an Oblique Triangle (Given SAS)

Area of a Triangle A B C c a b h Area = ½ ab(sin C) = ½ ac(sin B) = ½ bc (sin A)

Example – Finding the Area of a Triangular Lot Find the area of a triangular lot having two sides of lengths 90 meters and 52 meters and an included angle of 102 . Solution: Consider a = 90 meters, b = 52 meters, and the included angle C = 102  Then, the area of the triangle is Area = ½ ab sin C = ½ (90)(52)(sin102  )  2289 square meters.

20 Try It Out Determine the area of these triangles 127° ° 42.8° 17.9