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Hprec10 1

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  • 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. 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. 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. 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. Analysis A B C c b a xa-x h 2 2 2 b x h= + 2 2 2 ( )c a x h= − +
  • 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. 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. 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. 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. Example 100°60’ 90’ c 2 2 2 2 cosc a b ab C= + − 2 2 2 60 90 (60)(90)cos100c = + − °
  • 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. Example Solve triangle ABC (find the missing sides & angles) if A=39.4°,b=12 & c=14
  • 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. Example Solve triangle PQR if p=14, q=15 & r=16. Note: we are given 3 sides and asked to find 3 angles.
  • 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. 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. 319 mi 75 m i. d 15°
  • 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