hprec10 1
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10-1: The Law of CosinesLearning Goals:
©2008 Roy L. Gover (www.mrgover.com)
•Derive The Law of Cosines
•Solve triangles using The Law of Cosines
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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
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Important Idea
The Law of Cosines extends right triangle trigonometry to the measurement of sides and angles of triangles that are not right triangles.
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DefinitionLaw 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
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Analysis A
B C
c b
axa-x
h
2 2 2b x h 2 2 2( )c a x h
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A
B C
c b
axa-x
h
2 2 2( )c a x h 2 2 2 22c a ax x h
2 2 2b x h
2 2 22c a ax b
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A
B C
c b
axa-x
h
2 2 2( )c a x h 2 2 2 22c a ax x h 2 2 22c a ax b
cosx
Cb
cosx b C
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A
B C
c b
axa-x
h
2 2 2( )c a x h 2 2 2 22c a ax x h 2 2 22c a ax b
2 2 22 cosc a ab C b 2 2 2 2 cosc a b ab C
aka the Law of Cosines
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ExampleYou are building a house. The roof has the dimensions in the diagram. Find the width, c, of the lower beam.
100°
60’
90’
c
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Example
100°
60’
90’
c
2 2 2 2 cosc a b ab C 2 2 260 90 (60)(90)cos100c
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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.
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Example
Solve triangle ABC (find the missing sides & angles) if A=39.4°,b=12 & c=14
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Try ThisSolve 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’
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ExampleSolve triangle PQR if p=14, q=15 & r=16.
Note: we are given 3 sides and asked to find 3 angles.
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Try ThisSolve 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
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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?
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319 mi
75 mi.
d
15°
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Lesson CloseThe Law of Cosines is used to solve triangles when you know either:•Two sides and an included angle (SAS)
•Three sides (SSS)or