Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CHAPTER 8: Applications of Trigonometry 8.1 8.2 8.3 8.4 8.5 8.6 The Law of Sines The Law of Cosines Complex Numbers: Trigonometric Form Polar Coordinates and Graphs Vectors and Applications Vector Operations Copyright © 2009 Pearson Education, Inc. 8.2 The Law of Cosines Use the law of cosines to solve triangles. Determine whether the law of sines or the law of cosines should be applied to solve a triangle. Copyright © 2009 Pearson Education, Inc. Law of Cosines The Law of Cosines In any triangle ABC, B a b c 2bc cos A 2 2 2 b a c 2ac cos B 2 2 c 2 c 2 a 2 b 2 2ab cosC a A b C Thus, in any triangle, the square of a side is the sum of the squares of the other two sides, minus twice the product of those sides and the cosine of the included angle. When the included angle is 90º, the law of cosines reduces to the Pythagorean theorem. Copyright © 2009 Pearson Education, Inc. Slide 8.2 - 4 When to use the Law of Cosines The Law of Cosines is used to solve triangles given two sides and the included angle (SAS) or given three sides (SSS). Copyright © 2009 Pearson Education, Inc. Slide 8.2 - 5 Example In !ABC, a = 32, c = 48, and B = 125.2º. Solve the triangle. Solution: Draw and label a triangle. A? a 32 B 125.2º b ? C ? c 48 Copyright © 2009 Pearson Education, Inc. Slide 8.2 - 6 Example Solution continued Use the law of cosines to find the third side, b. b2 a2 c2 2accos B b2 322 482 2(32)(48)cos125.2º b2 5098.8 b 71 We need to find the other two angle measures. We can use either the law of sines or law of cosines. Using the law of cosines avoids the possibility of the ambiguous case. So use the law of cosines. Copyright © 2009 Pearson Education, Inc. Slide 8.2 - 7 Example Solution continued Find angle A. a2 b2 c2 2bccos A 32 71 48 2 71 48cos A 1024 5041 2304 6816 cos A 2 2 2 6321 6816cos A cos A 0.9273768 A 22.0º Now find angle C. C ≈ 180º – (125.2º + 22º) C ≈ 32.8º Copyright © 2009 Pearson Education, Inc. Thus, A 22.0º B 125.2º a 32 b 71 C 32.8º c 48 Slide 8.2 - 8 Example Solve !RST, r = 3.5, s = 4.7, and t = 2.8. Solution: Draw and label a triangle. R? S? T ? r 3.5 s 4.7 t 2.8 s 2 r 2 t 2 2rt cosS 4.7 3.5 2.8 2 2 Copyright © 2009 Pearson Education, Inc. 2 2 3.5 2.8 cosS Slide 8.2 - 9 Example Solution continued 2 2 2 3.5 2.8 4.7 cos S 2 3.5 2.8 cosS 0.1020408 S 95.86º Similarly, find angle R. r 2 s 2 t 2 2st cos R 3.5 4.7 2.8 2 4.7 2.8 cos R 2 2 2 4.7 2.8 3.5 cos S 2 4.7 2.8 2 2 Copyright © 2009 Pearson Education, Inc. 2 Slide 8.2 - 10 Example Solution continued cos R 0.6717325 R 47.80º Now find angle T. T ≈ 180º – (95.86º + 47.80º) ≈ 36.34º Thus, R 47.80º S 95.86º T 36.34º Copyright © 2009 Pearson Education, Inc. r 3.5 s 4.7 t 2.8 Slide 8.2 - 11 Example Knife makers know that the bevel of the blade (the angle formed at the cutting edge of the blade) determines the cutting characteristics of the knife. A small bevel like that of a straight razor makes for a keen edge, but is impractical for heavy-duty cutting because the edge dulls quickly and is prone to chipping. A large bevel is suitable for heavy-duty work like chopping wood. Survival knives, being universal in application, are a compromise between small and large bevels. The diagram illustrates the blade of a hand-made Randall Model 18 survival knife. What is its bevel? Copyright © 2009 Pearson Education, Inc. Slide 8.2 - 12 Example Solution: Use the law of cosines to find angle A. a2 b2 c2 2bccos A 0.5 2 2 2 2 2 2 2 2 cos A 4 4 0.25 cos A 8 cos A 0.96875 A 14.36º The bevel is approximately 14.36º. Copyright © 2009 Pearson Education, Inc. Slide 8.2 - 13