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10.2 The Law of Sines Objectives: 1. 2. Solve oblique triangles by using the Law of Sines. Use area formulas to find areas of triangles. The Law of Sines Solve the triangle. sin 115 sin 27 45 b b sin 115 45 sin 27 45 sin 27 b 22.5 sin 115 C 180 115 27 38 Ex. #1 sin 115 sin 38 45 c c sin 115 45 sin 38 45 sin 38 c 30.6 sin 115 Solve a Triangle with AAS Information For all these triangles, side a is called the swinging side and side b is called the fixed side. Ambiguous Case: SSA Ambiguous Case: SSA Given a possible triangle ABC with b=8, c=5, and C=54°, find angle B. First we draw the triangle. Since the exact shape is unknown, we start with a baseline and estimate a 54° angle. Side c must be across from angle C, so we place side b = 8 adjacent to the angle and c = 5 across from it. Since the angle is unknown, we can attach side c to b, we just don’t know the angle in which they meet. Side c in this scenario is called the swinging side. Ex. #2 Solve a Triangle with SSA Information Given a possible triangle ABC with b=8, c=5, and C=54°, find angle B. The swinging side can form any angle in a circular motion attached at point A. Depending on the height h, from A to the baseline, the swinging side could either form one triangle, two triangles, or no triangle. Ex. #2 Solve a Triangle with SSA Information Given a possible triangle ABC with b=8, c=5, and C=54°, find angle B. To find the height h, we will use right triangle trigonometry: h sin 54 8 h 8 sin 54 h 6. 5 The height is longer than the swinging side, which cannot reach the baseline to form a triangle. Ex. #2 Solve a Triangle with SSA Information Lighthouse B is 3 miles east of lighthouse A. Boat C leaves lighthouse B and sails in a straight line. At the moment that the boat is 5 miles from lighthouse B, an observer at lighthouse A notes that the angle determined by the boat, lighthouse A (the vertex), and lighthouse B is 65°. Approximately how far is the boat from lighthouse A at that moment? In this situation, the swinging side is now a = 5 as angle A is given. Because the swinging side is longer than the fixed side c = 3, it will automatically be longer than the height h and form exactly one triangle. h Ex. #3 Solve a Triangle with SSA Information Approximately how far is the boat from lighthouse A at that moment? To find side b, we must first find angle B. In order to find angle B we must first find angle C. sin 65 sin C 5 3 3 sin 65 5 sin C 3 sin 65 sin C 5 3 sin 65 C sin 1 5 C 32.94 Ex. #3 Solve a Triangle with SSA Information Approximately how far is the boat from lighthouse A at that moment? B 180 65 32.94 82.06 sin 82.06 sin 65 b 5 b sin 65 5 sin 82.06 5 sin 82.06 b sin 65 b 5.5 miles Ex. #3 Solve a Triangle with SSA Information Solve triangle ABC when a = 4.8, b = 6, & A = 28°. Here we must again first find the height in order to see if the swinging side a = 4.8 will reach down and touch the baseline to form a triangle. h sin 28 6 h 6 sin 28 h 2.8 Ex. #4 Solve a Triangle with SSA Information Solve triangle ABC when a = 4.8, b = 6, & A = 28°. With a height of 2.8, the swinging side is long enough to touch the baseline, but instead of forming just one triangle, it forms two triangles. 2.8 This occurs whenever the swinging side is longer than the height but shorter than the fixed side. Ex. #4 Solve a Triangle with SSA Information Solve triangle ABC when a = 4.8, b = 6, & A = 28°. Since two possible triangles are formed, both need solved. The small numbers next to the sides and angles indicate which triangle we are solving. sin 28 sin B1 4.8 6 4.8 sin B1 6 sin 28 6 sin 28 4.8 6 sin 28 B1 sin 1 4.8 B1 35.93 sin B1 Ex. #4 C1 180 28 35.93 116.07 sin 28 sin 116.07 4.8 c1 c1 sin 28 4.8 sin 116.07 4.8 sin 116.07 sin 28 c1 9.2 c1 Solve a Triangle with SSA Information Solve triangle ABC when a = 4.8, b = 6, & A = 28°. To solve the second triangle, we must first use a little geometry and our answer for angle B1 to find angle B2. Since both legs (formed by the swinging side) are the same length, they form an isosceles triangle. This means the two interior angles are congruent as well. Since we found B1 to be 35.93°, to find B2 we subtract it from 180°. B2 180 35.93 144.07 Ex. #4 Solve a Triangle with SSA Information Solve triangle ABC when a = 4.8, b = 6, & A = 28°. To find angle C2 we simplify must subtract angles A and B2 from 180°. C2 180 28 144.07 7.93 Finally we can solve for side c2. sin 28 sin 7.93 4.8 c2 c2 sin 28 4.8 sin 7.93 4.8 sin 7.93 sin 28 c2 1.4 c2 Ex. #4 Triangle 1 : B1 35.93 Triangle 2 : B2 144 .07 C1 116 .07 C2 7.93 c1 9.2 c2 1.4 Solve a Triangle with SSA Information Two surveyors, standing at points A and B, are measuring a building. The surveyor at point A is 20 feet further away from the building than the surveyor at point B. The angle of elevation of the top of the building from point A is 58°, and the angle of elevation of the top of the building from point B is 72°. How tall is the building? In order to figure out the height of the building, we will first need to find side a. Ex. #5 Solve a Triangle with ASA Information How tall is the building? To find side a, we will need the other angles in its triangle. 180 72 108 180 58 108 14 14° 108° Ex. #5 sin 14 sin 58 20 a a sin 14 20 sin 58 20 sin 58 a sin 14 a 70.11 Solve a Triangle with ASA Information How tall is the building? h sin 72 a 14° 108° Ex. #5 To find the height of the building h, we will use simple right triangle trigonometry. h sin 72 70.11 h 70.11sin 72 h 66.7 ft Solve a Triangle with ASA Information Caution! This formula can only be directly used when given SAS information. Area of a Triangle Given SAS Find the area of the triangle shown in the figure below: 1 A ab sin C 2 1 56 sin 153 2 15 sin 153 6.8 cm2 Ex. #6 Find Area with SAS Information Heron’s Formula: Use this formula when all sides of a triangle are known. The variable s represents the semi-perimeter which is half the perimeter of the triangle. Area of a Triangle Given SSS Find the area of the triangle whose sides have lengths of 8, 10, and 14. First find the semi-perimeter: s 1 a b c 2 1 8 10 14 2 1 32 16 2 Ex. #7 Then plug into Heron’s formula and simplify: A ss a s b s c 1616 816 10 16 14 1686 2 1536 39.2 square units Find Area with SSS Information