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• Work out problems on board • Add visuals (ranges of arccos and arcsin) to show why you use LOC for big angles and LOS for small angles 6.2 The Law of Cosines Which proved triangles congruent in Geometry? • • • • • • SSS ASA AAS SAS AAA ASS The same ones that define a specific triangle! • • • • SSS - congruent ASA - congruent AAS – congruent SAS – congruent • AAA – not congruent • ASS – not congruent Which proved triangles congruent in Geometry? • • • • SSS - congruent ASA – congruent – Solve w/ Law of Sines AAS – congruent – Solve w/ Law of Sines SAS – congruent • AAA – not congruent • ASS – not congruent 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? 15 No side opposite from any angle to get the ratio 26° 12.5 6 Let's consider types of triangles with the three pieces of information shown below. We can't use the Law of Sines on these because we don't have an angle and a side opposite it. We need another method for SAS and SSS triangles. SAS You may have a side, an angle, and then another side SSS You may have all three sides AAA You may have all three angles. AAA This case doesn't determine a triangle because similar triangles have the same angles and shape but "blown up" or "shrunk down" Do you see a pattern? LAW OF COSINES c 2 a 2 b 2 2ab cos C b 2 a 2 c 2 2ac cos B a 2 b 2 c 2 2bc cos A Deriving the Law of Cosines • Write an equation using Pythagorean theorem for shaded triangle that only includes sides and angles of the oblique triangle. C b h k a c-k A a h (c k ) 2 h b sin A k b cos A 2 c 2 a b sin A c b cos A 2 B 2 2 a 2 b 2 sin 2 A c 2 2 c b cos A b 2 cos 2 A a 2 b 2 sin 2 A cos 2 A c 2 2 c b cos A a 2 b 2 c 2 2 c b cos A 9 Since the Law of Cosines is more involved than the Law of Sines, when you see a triangle to solve you first look to see if you have an angle (or can find one) and a side opposite it. You can do this for ASA, AAS and SSA. In these cases you'd solve using the Law of Sines. However, if the 3 pieces of info you know don't include an angle and side opposite it, you must use the Law of Cosines. These would be for SAS and SSS (remember you can't solve for AAA). Ex. 1: Solve a triangle where b = 1, c = 3 and A = 80° Draw a picture. This is SAS Do we know an angle and side opposite it? No so we must use Law of Cosines. Hint: we will be solving for the side opposite the angle we know. One side squared Now punch buttons on your calculator to find a. It will be square root of right hand side. a = 2.9930 B 3 a C 80 1 a b c 2bc cos A 2 2 2 sum of each of the other sides squared minus 2 times the times the cosine of product the angle of those between other those sides sides a 1 3 213cos 80 2 2 2 CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction We'll label side a with the value we found. We now have all of the sides but how can we find an angle? 3 80 Hint: We have an angle and a side opposite it. sin 80 sin B 2.993 1 sin 80 sin B 2.993 B 19.21 B 19.21 2.993 C 80.79 1 C is easy to find since the sum of the angles is a triangle is 180° 180 80 19.21 80.79 When taking arcsin, use 2nd answer on your calculator for accuracy! Ex. 2: Solve a triangle where a = 5, b = 8 and c = 9 Draw a picture. This is SSS 9 Do we know an angle and side opposite it? No, so we must use Law of Cosines. Let's choose to find angle C first. B A 5 84.26 C 8 c a b 2ab cos C 2 One side squared 2 2 sum of each of the other sides squared minus 2 times the times the cosine of product the angle of those between other those sides sides 81 89 80 cos C 1 1 8 C cos 10 84.2608 cos C 2 2 2 80 258cos C 9 5 8 CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction How can we find one of the remaining angles? Do we know an angle and side opposite it? 9 B 62.18 5 84.26 33.56 A Yes, so use Law of Sines. 8 sin 84.26 sin B 9 8 8 sin 84.2608 sin Bsin 1 8 sin 84.2608 62.1819 9 9 A 180 84.2608 62.1819 33.5573 Too easy, what’s the catch? • After we use L.O.C. we need to use law of sines to find the remaining sides and angles. • The range of arcsin is -90 deg to 90 deg, but what if the angle is obtuse? Then taking the arcsin won’t get us the correct angle! • To avoid this problem – When using L.O.S. after L.O.C. always find the smallest angle FIRST The smallest angle has to be acute since there can’t be more than one obtuse angle in a triangle. • Then use the triangle sum thm to find the 3rd angle. Try it on your own! #1 • Find the three angles of the triangle ABC if a 6, b 8, c 12 C 8 A 6 12 B A 26.38 , B 36.34 , C 117.28 16 Try it on your own! #2 • Find the remaining angles and side of the triangle ABC if b 16, c 12, m A 80 C 16 A 80 B 12 a 18.257, B 59.67 , C 40.33 17 Summary – What could we use to solve the following triangles? 70 80 30 Uh, nothing. It’s AAA 18 Summary – What could we use to solve the following triangles? 20 16 80 ASA – although we don’t know an angle and side opposite each other we can find the 3rd angle then do law of sines 19 Summary – What could we use to solve the following triangles? 16 20 80 AAS – law of sines 20 Summary – What could we use to solve the following triangles? 16 20 80 ASS, we can use law of sines but need to check for 1, 2 or no triangles. 21 Summary – What could we use to solve the following triangles? 16 80 12 SAS – don’t know (and can’t find) angle and side opposite Law of Cosines 22 Summary – What could we use to solve the following triangles? 16 20 12 SSS – don’t know (and can’t find) angle and side opposite Law of Cosines 23 Wing Span C • The leading edge of each wing of the B-2 Stealth Bomber A measures 105.6 feet in length. The angle between the wing's leading edges is 109.05°. What is the wing span (the distance from A to C)? • Note these are the actual dimensions! 24 Wing Span C b 2 a 2 c 2 2ac cos B A b 105.6 105.6 2(105.6)(105.6) cos109.05 2 2 2 b 22302.72 7279.46 2 b 172 ft. 25 • The direction to a point is stated as the number of degrees east or west of north or south. For example, the direction of • A from O is N30ºE. B is N60ºW from O. C is S70ºE from O. D is S80ºW from O Navigational Bearings H Dub • 6-2 Pg. 443 #2-16even, 17-22all, 29, 34, 35 Practice #1 Practice #2 Do you see a pattern? LAW OF COSINES Use these to find missing sides c a b 2ab cos C 2 2 2 b 2 a 2 c 2 2ac cos B a 2 b 2 c 2 2bc cos A LAW OF COSINES b c a cos B a c b cos A 2ac 2bc 2 2 2 a b c Use these to find cos C missing angles 2ab 2 2 2 2 2 2 Practice #1 Practice #2