Chapter 2 Trigonometry 2.4 The Cosine Law Explain why the sine law cannot be used to solve each triangle. A B 28° 37° 115° C J K L 70 40 45 There is no known side opposite a known angle. There is no known angle opposite a known side. P Q R 15 E D 85° 20 25 F There is no known angle and only one known side. There is no known angle opposite a known side Pre-Calculus 11

a2 = b2 + c2 -2bc cosA b2 = a2 + c2 -2ac cosB c2 = a2 + b2 -2ab cosC Law of Cosines a2 = b2 + c2 -2bc cosA b2 = a2 + c2 -2ac cosB c2 = a2 + b2 -2ab cosC When should we use the Law of Cosine to solve a triangle? SSS Given the measure of three sides. SAS Given the measure of two sides and one angle not opposite a side. Pre-Calculus 11

Proof for the Law of Cosines B C A a c pythagorean h x D b - x sohcahtoa b Pre-Calculus 11

b2 = a2 + c2 -2ac cosB a2 = b2 + c2 -2bc cosA a = 37.9 cm SAS Applying the Law of Cosines (to the nearest 10th) Determine the length of side b b2 = a2 + c2 -2ac cosB = (230)2 + (150)2 - 2(230)(150)cos430 b = 157.9 m Determine the length of side a a2 = b2 + c2 -2bc cosA = (61)2 + (43)2 - 2(61)(43)cos380 a = 37.9 cm Pre-Calculus 11

SSS Finding an Angle Using the Law of Cosines (to the nearest degree) Determine the measure of angle A. a2 = b2 + c2 -2bc cosA 382 = 612 + 432 -2(61)(43) cosA 382 - 612 - 432 = -2(61)(43) cosA Pre-Calculus 11

SSS Finding an Angle Using the Law of Cosines (to the nearest degree) Given triangle DEF, find Pre-Calculus 11

h Solving Two Triangles b2 = (95)2 + (200)2 - 2(95)(200)cos50 From the following information determine the height of the balloon from the ground (to the nearest tenth) b2 = (95)2 + (200)2 - 2(95)(200)cos50 b2 ~ 24599.07083 b = 156.841 (we need length side b before we can calculate h) ← Sine Law h h = 263. 7 The balloon is 263.7 m from the ground b Pre-Calculus 11

Begin by using the method Solving Triangles When solving triangles, it is important to choose the most appropriate method. The choice depends on the given information. Place the letter of the appropriate method beside the given information. Given Information Begin by using the method Three sides Three angles Two angles and any side Right triangle Two sides and the angle between them Two sides and the angle opposite one of the sides A. Primary trig ratio SSS C AAA D B. sine law AAS B C. cosine law A D. none of the above SAS C SSA B Pre-Calculus 11

Assignment Suggested Questions Page 119: (1-6)ac, 8, 10, 12, 14, 17, 19, 20, 23 Pre-Calculus 11