Transcript Sullivan Algebra and Trigonometry: Section 9.3

Sullivan PreCalculus
Section 7.3
The Law of Cosines
Objectives of this Section
• Solve SAS Triangles
• Solve SSS Triangles
• Solve Applied Problems
We use the Law of Sines to solve CASE 1
(SAA or ASA) and CASE 2 (SSA) of an
oblique triangle. The Law of Cosines is
used to solve CASES 3 and 4.
CASE 3: Two sides and the included
angle are known (SAS).
CASE 4: Three sides are known (SSS).
Law of Cosines
For a triangle with sides a , b, c and opposite
angles  ,  , , respectively.
c  a  b  2ab cos
2
2
2
b  a  c  2ac cos 
2
2
2
a  b  c  2bc cos
2
2
2
Solve the triangle: b  3, c  4,  40 (SAS)

3
a

40
4
a  b  c  2bc cos
2
2
2
a  3  4  234 cos 40
2
2
2
a  6.614933365
a  2 . 57
2

3
2.57

40
4
2
2
2
b  a  c  2ac cos 
a c b
6.6149  4  3
cos  

 0.6622
2ac
2(2.57)(4)

  48.5
2
2
2
  180  40  48.5
2
2
  915
.
Solve the triangle: a  3, b  5, c  7 (SSS)

5
3


a  b  c  2bc cos
2
2
2
b c a
cos 
2bc
2
2
2
7
5 7 3
cos 
2(5)(7)
65 13
cos 

70 14
2
  21.8
2

2

5
3


7
2
2
2
a  c b
3 7 5
33 11
cos  



2ac
2( 3)(7)
42 42
2
2
2
  74.8

  180  218
.  74.8  83.4