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Objectives: 1. To derive and use the Law of Cosines • • • • Assignment: P. 443: 1-16 S P. 443: 17-22 S P. 443-445: 29-43 S HW Supplement You will be able to derive and use the Law of Cosines Find the length of AC in acute triangle ABC. As the previous example demonstrates, you cannot always use the Law of Sines for every triangle. You need either two sides and an angle or two angles and a side in the following configurations. What we need is another law. AAS SSA • Ambiguous On the same set of trigonometric axis, graph the following functions. y cos x y cos x In the up-coming activity, the identity below will prove tremendously useful. cos x cos x In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. c2 a2 b2 If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then it is a right triangle. Tell whether a triangle with the given side lengths is a right triangle. 1. 5, 6, 7 2. 5, 6, 61 3. 5, 6, 8 In the previous example, the side lengths 5 and 6 do not make a right triangle when paired with either 7 or 8. So what kind of triangle do these side lengths make? You can use the Pythagorean formula to determine whether a triangle is right or not: Acute: c a b 2 2 2 Need to subtract something to = c2 Right: c a b 2 Where c is the longest side. 2 2 Obtuse: c a b 2 2 2 Need to add something to = c2 To derive the Law of Cosines, we need an interesting diagram like the one shown. Click the picture for a Geometer’s Sketchpad Demonstration of this useful equation. If ΔABC has sides of length a, b, and c as shown, then c 2 a 2 b 2 2ab cos C b a c 2ac cos B 2 2 2 a 2 b 2 c 2 2bc cos A Simplify the equation below for C = 90°. c 2 a 2 b 2 2ab cos C When C is acute, cos C is positive: Acute: c a b 2 2 2 Need to subtract something to = c2 c 2 a 2 b 2 2ab cos C When C is obtuse, cos C is negative: Obtuse: c a b 2 2 2 Need to add something to = c2 c 2 a 2 b 2 2ab cos C This becomes minus a negative, so you are actually adding something Solve ΔABC. Solve ΔQED. Solve the equation below for C. c 2 a 2 b 2 2ab cos C Solve ΔABC. When using the Law of Cosines to find a missing 1: Use the Law of Cosines angle (SSS), it’s a good idea to find the angle opposite the longest side first. This is just in case the angle turns out to be obtuse. Regardless of what type of angle this turns out to be, use the 2: Use the Law of Sines Law of Sines and the Triangle Sum Theorem to 3: Use the Triangle Sum find the other two angles. Find the indicated measure. 1. x and y 2. and Law of Sines • ASA • AAS • SSA Law of Cosines • SAS • SSS Given three pieces of any triangle, you can use the Law of Sines or the Law of Cosines to completely solve the triangle. A sailboat leaves its pier at a bearing of 270 at 8 knots (nautical miles per hour). After 15 minutes, the boat changes course to a bearing of 344 at 10 knots. Find the sailboat’s bearing and distance from the pier after 12 minutes on this course. Armed with a compass and an odometer, you set off on a bike trail consisting of two legs. The first leg takes you 20 miles at a bearing of 050, while on the second leg, you ride 12 miles at a bearing of 125. What is the distance, as the crow flies, from the starting point to your destination? What is the bearing? A ship travels 40 nm due east and then changes direction. When the ship has traveled 30 nm at this heading, it is 56 nm from its point of departure. Describe the bearing from B to C. Objectives: 1. To derive and use the Law of Cosines • • • • Assignment: P. 443: 1-16 S P. 443: 17-22 S P. 443-445: 29-43 S HW Supplement