10-1: The Law of Cosines
Learning Goals:
©2008 Roy L. Gover (www.mrgover.com)
•Derive The Law of Cosines
•Solve triangles ...
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
Important Idea
The Law of Cosines
extends right triangle
trigonometry to the
measurement of sides and
angles of triangles ...
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 ...
Analysis A
B C
c b
a
xa-x
h
2 2 2
b x h= +
2 2 2
( )c a x h= − +
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= − +
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=
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
...
Example
You are building a house.
The roof has the dimensions
in the diagram. Find the
width, c, of the lower beam.
100°60...
Example
100°60’ 90’
c
2 2 2
2 cosc a b ab C= + −
2 2 2
60 90 (60)(90)cos100c = + − °
Important Idea
•Don’t forget to set the
mode on your calculator.
•Throughout this section,
no rounding is done in
the comp...
Example
Solve triangle ABC (find the
missing sides & angles) if
A=39.4°,b=12 & c=14
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 n...
Example
Solve triangle PQR if p=14,
q=15 & r=16.
Note: we are given 3
sides and asked to find 3
angles.
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° ...
Example
•Natalie is
flying from
Dallas to
Little Rock, a distance of 319
miles. She starts her flight
15° off course and f...
319 mi
75
m
i.
d
15°
Lesson Close
The Law of Cosines is used
to solve triangles when you
know either:
•Two sides and an included
angle (SAS)
•T...
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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
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