Hprec10 1

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

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 that are not right triangles.

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

Analysis A

B C

c b

axa-x

h

2 2 2b x h 2 2 2( )c a x h

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

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

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

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

Example

100°

60’

90’

c

2 2 2 2 cosc a b ab C 2 2 260 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 computation. Round only the final answer.

Example

Solve triangle ABC (find the missing sides & angles) if A=39.4°,b=12 & c=14

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’

ExampleSolve triangle PQR if p=14, q=15 & r=16.

Note: we are given 3 sides and asked to find 3 angles.

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

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?

319 mi

75 mi.

d

15°

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