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23-3 and 23-4: The Law of Cosines Objectives: 1. To derive and use the Laws of Cosines Assignment: • P. 329-330: 9-15 • Challenge Problems Objective 1 You will be able to derive and use the Law of Cosines Example 1 Find the length of 𝐴𝐶 in acute triangle 𝐴𝐵𝐶. Law of Sines 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: ASA AAS SSA Law of Cosines 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. Law of Cosines If 𝐴𝐵𝐶 has sides of length a, b, and c as shown, then 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 Example 2 Simplify the equation below for C = 90°. c 2 a 2 b 2 2ab cos C General Pythagorean Theorem When 𝐶 is acute, cos 𝐶 is positive: Acute: c2 a 2 b2 Need to subtract something to = c2 c 2 a 2 b 2 2ab cos C General Pythagorean Theorem When 𝐶 is obtuse, cos 𝐶 is negative: Obtuse: c2 a 2 b2 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 Example 3 Solve the equation below for C. c 2 a 2 b 2 2ab cos C Example 1 (Revisited) Find the length of 𝐴𝐶 in acute triangle 𝐴𝐵𝐶. Example 4 Solve Δ𝐴𝐵𝐶. Pro Tip: SSS When using the Law of Cosines to find a missing 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 Law of Sines and the Triangle Sum Theorem to find the other two angles. 1: Use the Law of Cosines 2: Use the Law of Sines 3: Use the Triangle Sum Example 5 Find the indicated measure. 1. x and y 2. and Summary Law of Sines Law of Cosines • ASA • AAS • SSA • 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. Example 6 Given that 𝑚∠𝐵𝐴𝐷 = 101°, find the lengths of 𝐴𝐵, 𝐴𝐵, and 𝐴𝐶. C 11.11 ft B 7.44 ft A D 23-3 and 23-4: The Law of Cosines Objectives: 1. To derive and use the Laws of Cosines Assignment: • P. 329-330: 9-15 • Challenge Problems