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10-1: The Law of Cosines

10-1: The Law of Cosines. Essential Question: What is the law of cosines, and when do we use it?. 10-1: The Law of Cosines. In any triangle ABC, with side lengths a, b, c – which are opposite their respective angle, the Law of Cosines states: a 2 = b 2 + c 2 – 2bc cos A

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10-1: The Law of Cosines

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  1. 10-1: The Law of Cosines Essential Question: What is the law of cosines,and when do we use it?

  2. 10-1: The Law of Cosines • In any triangle ABC, with side lengths a, b, c – which are opposite their respective angle, the Law of Cosines states: • a2 = b2 + c2 – 2bc cos A • b2 = a2 + c2 – 2ac cos B • c2 = a2 + b2 – 2ab cos C • Basically: It’s just like the Pythagorean Theorem, then subtracting two times those two sides times the cosine of the angle. • Proof on board

  3. 10-1: The Law of Cosines • The law of Cosines can be used to solve triangles in the following cases: • Given two sides and an angle between them (SAS) • Given three sides (SSS) • The law of cosines helps us solve the situations where the law of sines cannot.

  4. 10-1: The Law of Cosines • Example 1: Solve a Triangle with SAS Information. • Solve triangle ABC below 1) Use law of cosines to find c c2 = a2 + b2 – 2ab cos C c2 = 162 + 102 – 2(16)(10) cos 110 c2 = 256 + 100 – 320(-0.3420) c2 = 356 + 109.4464 c2 = 465.4464 c  21.5742 (you can give 21.6 as an answer, but use 4 digits to continue solving) C 2) Use law of sines to find A (or B) 110° 10 16 3) Find the last angle A B B = 180 – 110 – 44.2 = 25.8 c

  5. 10-1: The Law of Cosines • Example 2: Solve a Triangle with SSS Information • Solve a triangle where a = 20, b = 15 and c = 8.3 • Use the law of cosines to find any angle • Use the law of sines to find another angle • Use common sense to find the third angle c2 = a2 + b2 – 2ab cos C -556.11 = -600 cos C 8.32 = 202 + 152 – 2(20)(15) cos C .92685 = cos C 68.89 = 400 + 225 – 600 cos C cos-1(.92685) = C 68.89 = 625 – 600 cos C 22.05° = C C = 180 – 22.1 – 42.7 C = 115.2

  6. 10-1: The Law of Cosines • Example 3: The distance between two vehicles • Two trains leave a station on different tracks. The tracks make an angle of 125° with the station as the vertex. The first train travels at an average speed of 100 km/h, and the second train travels at an average speed of 65 km/h. How far apart are the trains after 2 hours? • These questions are helped if you draw a diagram. 1st x 130 125° 2nd 200 Station

  7. 10-1: The Law of Cosines • Example 3: The distance between two vehicles • Use the law of cosines • x2 = 1302 + 2002 – 2(100)(200) cos 125 • x2 = 16900 + 40000 – 52000 cos 125 • x2 = 86725.975 • x = 294.5 1st x 130 125° 2nd 200 Station

  8. 10-1: The Law of Cosines • Assignment • Page 622 • Problems 1 – 25, odds • Show work

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