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# Hprec10 1

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### Hprec10 1

1. 1. 10-1: The Law of Cosines Learning Goals: ©2008 Roy L. Gover (www.mrgover.com) •Derive The Law of Cosines •Solve triangles using The Law of Cosines
2. 2. Try This A B C 7 49.2° b c Solve triangle ABC. Round angle and side measures to the nearest tenth. B=40.8°, b=6.0,c=9.2
3. 3. Important Idea The Law of Cosines extends right triangle trigonometry to the measurement of sides and angles of triangles that are not right triangles.
4. 4. Definition Law of Cosines in 3 parts: 2 2 2 2 cosa b c bc A= + − 2 2 2 2 cosb a c ac B= + − 2 2 2 2 cosc a b ab C= + − Do you see a pattern? A CB a bc
5. 5. Analysis A B C c b a xa-x h 2 2 2 b x h= + 2 2 2 ( )c a x h= − +
6. 6. A B C c b a xa-x h 2 2 2 ( )c a x h= − + 2 2 2 2 2c a ax x h= − + + 2 2 2 b x h= + 2 2 2 2c a ax b= − +
7. 7. A B C c b a xa-x h 2 2 2 ( )c a x h= − + 2 2 2 2 2c a ax x h= − + + 2 2 2 2c a ax b= − + cos x C b = cosx b C=
8. 8. A B C c b a xa-x h 2 2 2 ( )c a x h= − + 2 2 2 2 2c a ax x h= − + + 2 2 2 2c a ax b= − + 2 2 2 2 cosc a ab C b= − + 2 2 2 2 cosc a b ab C= + − aka the Law of Cosines
9. 9. Example You are building a house. The roof has the dimensions in the diagram. Find the width, c, of the lower beam. 100°60’ 90’ c
10. 10. Example 100°60’ 90’ c 2 2 2 2 cosc a b ab C= + − 2 2 2 60 90 (60)(90)cos100c = + − °
11. 11. Important Idea •Don’t forget to set the mode on your calculator. •Throughout this section, no rounding is done in the computation. Round only the final answer.
12. 12. Example Solve triangle ABC (find the missing sides & angles) if A=39.4°,b=12 & c=14
13. 13. Try This Solve triangle ABC (find the missing sides & angles) if A=52°10’,b=6 & c=8.Round angle and side measures to the nearest tenth. a=6.4,B=46.8°’,C=80.1’
14. 14. Example Solve triangle PQR if p=14, q=15 & r=16. Note: we are given 3 sides and asked to find 3 angles.
15. 15. Try This Solve triangle ABC if a= 21,b=16.7 & c =10.3. Round angle measures to the nearest tenth. A=99.4° B=51.7° C=28.9° A B C 21 16.7 10.3
16. 16. Example •Natalie is flying from Dallas to Little Rock, a distance of 319 miles. She starts her flight 15° off course and flies for 75 miles. How far is she from Little Rock?
17. 17. 319 mi 75 m i. d 15°
18. 18. Lesson Close The Law of Cosines is used to solve triangles when you know either: •Two sides and an included angle (SAS) •Three sides (SSS) or