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Law of Cosines HOMEWORK: Lesson 12.4/1-14 Who's Law Is It, Anyway? Murphy's Law: Anything that can possibly go wrong, will go wrong (at the worst possible moment). Cole's Law ?? Finely chopped cabbage 2 Solving an SAS Triangle The Law of Sines was good for ASA AAS SSA - two angles and the included side - two angles and any side - two sides and an opposite angle (being aware of possible ambiguity) Why would the Law of Sines not work for an SAS triangle? No side opposite from any angle to get the ratio 15 26° 12.5 3 Law of Cosines Note the pattern a b c 2 c b cos A 2 2 2 b a c 2 a c cos B 2 2 2 c b a 2 a b cos C 2 2 2 C b A a c B 4 We could do the same thing if gamma was obtuse and we could repeat this process for each of the other sides. We end up with the following: LAW OF COSINES Use these to find missing sides c 2 a 2 b 2 2ab cos b a c 2ac cos 2 2 2 a 2 b 2 c 2 2bc cos LAW OF COSINES b c a cos a c b cos 2ac 2bc a 2 b2 c2 cos 2ab 2 Use these to find missing angles 2 2 2 2 2 Applying the Cosine Law Now use it to solve the triangle we started with C 15 Label sides and angles Side c first A 26° c 12.5 B c b a 2 a b cos C 2 2 2 c 152 12.52 2 15 12.5 cos 26 6 Applying the Cosine Law C 15 A 26° c = 6.65 12.5 B Now calculate the angles 2 2 2 b a c 2 a c cos B use and solve for B b2 a 2 c 2 cos B 2 a c 2 2 2 b a c 1 B cos 2 a c 7 Applying the Cosine Law C 15 A 26° c = 6.65 12.5 B The remaining angle determined by subtraction 180 – 93.75 – 26 = 60.25 8 Solve a triangle where b = 1, c = 3 and = 80° Draw a picture. This is SAS 3 a Hint: we will be solving for the side opposite the angle we know. 80 Do we know an angle and side opposite it? No so we must use Law of Cosines. 1 a b c 2bc cos 2 2 2 a 1 3 213cos 80 2 2 2 a = 2.99 Solve a triangle where a = 5, b = 8 and c = 9 This is SSS Draw a picture. 9 5 Do we know an angle and side opposite it? No, so we must use Law of Cosines. 8 Let's use largest side to find largest angle first. 9 5 2 84.3 c a b 2ab cos 2 2 2 2 cos 2 5 8 8 2 81 89 80 cos 8 1 1 cos cos 84.3 80 10 9 5 84.3 8 11 Wing Span C The leading edge of each wing of the B-2 Stealth Bomber measures 105.6 feet A in length. The angle between the wing's leading edges is 109.05°. What is the wing span (the distance from A to C)? Hint … use the law of cosines! 12 C b 2 a 2 c 2 2 a c cos B x B 109.05° A 13 Using the Cosine Law to Find Area Recall that We can use the value for h to determine the area 1 Area c b sin A 2 h b sin A C b h a A B c 14 Using the Cosine Law to Find Area We can find the area knowing two sides and the included angle C 1 Area a b sin C 2 1 c a sin B 2 b A a c B Note the pattern 15 Try It Out Determine the area 12m 127° 24m 16 Determine the area Missing angle – 180-42.8-76.3 = 60.9° 17.9 60.9° 42.8° Missing side sin 76.3 sin 60.9 17.9 a 17 Cost of a Lot An industrial piece of real estate is priced at $4.15 per square foot. Find, to the nearest $1000, the cost of a triangular lot measuring 324 feet by 516 feet by 412 feet. 516 18 516 19 We'll label side a with the value we found. We now have all of the sides but how can we find an angle? Hint: We have an angle and a side opposite it. 3 2.99 80.8 80 sin 80 sin 2.99 3 3sin 80 80.8 2.99 19.2 1 is easy to find since the sum of the angles is a triangle is 180° 180 80 80.8 19.2