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5.4 Law of Cosines

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ASA or AAS SSA What’s left??? Law of Cosines Law of Sines Which one to use?? whichever is appropriate Law of Sines (always thinking about “ambiguous case”) … SAS and SSS

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Ex 1) Solve △ ABC if m ∠ C = 79 °, a = 25 and b = 29 Have SAS Find c (Law of Cosines) 25 29 c B C A 79° 34.5 Now, m ∠ A (or m ∠ B … doesn’t really matter which one) A = 45.3° m ∠ B = 180 – 79 – 45.3 = 55.7° Use Law of Sines

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Ex 2) Solve △ ABC if a = 14.3, b = 10.6 and c = 8.4 Divide class into 3 different groups Each group solves the △ in a different order 14.3 10.6 8.4 B C A Group 1 1.m ∠ A (law of cos) 2.m ∠ B 3.m ∠ C law of sines Group 2 1.m ∠ B (law of cos) 2.m ∠ A 3.m ∠ C law of sines Group 3 1.m ∠ C (law of cos) 2.m ∠ A 3.m ∠ B law of sines

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Ex 2) Solve △ ABC if a = 14.3, b = 10.6 and c = 8.4 Each group will list their answers 14.3 10.6 8.4 B C A Group 1 1.m ∠ A = 96.9° 2.m ∠ B = 47.4° 3.m ∠ C = 35.7° Group 2 1.m ∠ A = 83.4° 2.m ∠ B = 47.4° 3.m ∠ C = 35.7° Group 3 1.m ∠ A = 83.4° 2.m ∠ B = 47.4° 3.m ∠ C = 35.7° WHAT!?! Who is right? Add them up! Total = 180° Total = 166.5° Only Group 1 is right… why? What is unique about their situation that made them get the right answers but the rest of us didn’t? So… if we have SSS, what’s the best order to solve them?

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“Heading” is used in navigation The degree measure is calculated differently than in the rest of math Math: start go counter-clockwise Heading: start (due N) go clockwise

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Ex 3) Two airplanes leave an airport at the same time. The heading of the first is 150º and the heading of the second is 260º. If the planes travel at the rates of 680 mi/h and 560 mi/h, respectively, how far apart are they after 2 hours? 150° 260° 80° b = 2036 mi 10° A Plane I (150°): Plane II (260°): 30° B C 110° 1360 1120 b

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Homework #504 Pg 269 #12, 14, 21, 24, 25, 27, 28, 29, 33, 35 Answers to Evens: 12)59.7 14)539.3 ft 24)3203 miles 28) 20 nautical miles #24 & 25: When a pilot changes the heading, redraw the axes at that location

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Aim: What is the law of cosine? Do Now: In AB = c, BC = a, and AC = b. y x A B(c,0) C(x,y)C(x,y) a b c 1. In Δ ACD, find cos A 2. In ΔACD, find sin A.

Aim: What is the law of cosine? Do Now: In AB = c, BC = a, and AC = b. y x A B(c,0) C(x,y)C(x,y) a b c 1. In Δ ACD, find cos A 2. In ΔACD, find sin A.

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