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The Law of Cosines The Law of Cosines If A, B, mid C are the measures of the angles of a triangle, and a, b, and c are the lengths of the sides opposite these angles, then a2 = b2 + c2 - 2bc cos A b2 = a2 + c2 - 2ac cos B c2 = a2 + b2 - 2ab cos C. The square of a side of a triangle equals the sum of the squares of the other two sides minus twice their product times the cosine of their included angle. Solving an SAS Triangle 1. Use the Law of Cosines to find the side opposite the given angle. 2. Use the Law of Sines to find the angle opposite the shorter of the two given sides. This angle is always acute. 3. Find the third angle. Subtract the measure of the given angle and the angle found in step 2 from 180º. Text Example C b = 20 • Solve the triangle shown with A = 60º, b = 20, and c = 30. A a 60º c =30 B Solution We are given two sides and an included angle. Therefore, we apply the three-step procedure for solving a SAS triangle. Step l Use the Law of Cosines to find the side opposite the given angle. Thus, we will find a. a2 = b2 + c2 - 2bc cos A a2 = 202 + 302 - 2(20)(30) cos 60º = 400 + 900 - 1200(0.5) = 700 a = 700 = 26 Apply the Law of Cosines to find a. b = 20, c = 30, and A = 60°. Perform the indicated operations. Take the square root of both sides and solve for a. Text Example cont. C b = 20 a Solve the triangle shown with A = 60º, b = 20, and c = 30. A 60º c = 30 Solution Step 2 Use the Law of Sines to find the angle opposite the shorter of the two given sides. This angle is always acute. The shorter of the two given sides is b = 20. Thus, we will find acute angle B. b a = sin B sin A 20 700 = sin B sin 60 700 sin B = 20sin 60 sin B = B B 41 41 20sin 60 0.6547 700 Apply the Law of Sines. We are given b = 20 and A = 60°. Use the exact value of a, 700, from step 1. Cross multiply. Divide by square root of 700 and solve for sin B. Find sin-10.6547 using a calculator. B Text Example cont. C Solve the triangle shown with A = 60º, b = 20, and c = 30. b = 20 A a 60º c = 30 Solution We are given two sides and an included angle. Therefore, we apply the three-step procedure for solving a SAS triangle. Step 3 Find the third angle. Subtract the measure of the given angle and the angle found in step 2 from 180º. C = 180º - A - B = 180º - 60º - 41º = 79º The solution is a = 26, B = 41º, and C = 79º. B Solving an SSS Triangle 1. Use the Law of Cosines to find the angle opposite the longest side. 2. Use the Law of Sines to find either of the two remaining acute angles. 3. Find the third angle. Subtract the measures of the angles found in steps 1 and 2 from 180º. Text Example • Two airplanes leave an airport at the same time on different runways. One flies at a bearing of N66ºW at 325 miles per hour. The other airplane flies at a bearing of S26ºW at 300 miles per hour. How far apart will the airplanes be after two hours? Solution After two hours. the plane flying at 325 miles per hour travels 325 · 2 miles, or 650 miles. Similarly, the plane flying at 300 miles per hour travels 600 miles. The situation is illustrated in the figure. Let b = the distance between the planes after two hours. We can use a north-south line to find angle B in triangle ABC. Thus, B = 180º - 66º - 26º = 88º. We now have a = 650, c = 600, and B = 88º. Text Example cont. Two airplanes leave an airport at the same time on different runways. One flies at a bearing of N66ºW at 325 miles per hour. The other airplane flies at a bearing of S26ºW at 300 miles per hour. How far apart will the airplanes be after two hours? Solution We use the Law of Cosines to find b in this SAS situation. b2 = a2 + c2 - 2ac cos B Apply the Law of Cosines. b2 = 6502 + 6002 - 2(650)(600) cos 88º Substitute: a= 650, c =600, and B= 88°. = 755,278 Use a calculator. b = 869 Take the square root and solve for b. After two hours, the planes are approximately 869 miles apart. Heron’s Formula The area of a triangle with sides a, b, and c is Area = s ( s - a )( s - b)( s - c) 1 s = (a + b + c) 2 Example • Use Heron’s formula to find the area of the given triangle: a=10m, b=8m, c=4m Solution: 1 s = (a + b + c) 2 1 s = (10 + 8 + 4) 2 1 s = (22) = 11 2 Area = s( s - a)( s - b)( s - c) = 11(11 - 10)(11 - 8)(11 - 4) = 11(1)(3)(7) = 231 sq.m.