3.2 The Law of Cosines
■ Solving SAS and SSS Triangles (Cases 3 and 4)
■ Formulas for the Area of a Triangle
Triangle Side Length Restriction
In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Law of Cosines
In any triangle ABC, with sides a, b, and c, the following hold.
Standard Forms:
Alternative Forms:
a  b  c  2bc cos A,
2
2
2
b 2  a 2  c 2  2ac cos B ,
c 2  a 2  b 2  2ab cos C
b2  c 2  a 2
cos A 
2bc
2
a  c 2  b2
cos B 
2ac
2
a  b2  c 2
cos C 
2ab
Solving SSS and SAS Triangles (Cases 3 and 4)
CLASSROOM EXAMPLE 1
(SSS) (Home: copy Text Ex. 1 page 291)
Solve triangle ABC if a = 25.4 cm, b = 42.8 cm, and c = 59.3 cm.
1) Find the angle opposite to the longest side using the Law of Cosines.
2) Use the Law of Sines to find another angle.
3) Find the third angle.
(SAS) (Home: copy Text Ex. 2 page 292)
Solve triangle ABC if B = 73.5°, a = 28.2 ft, and c = 46.7 ft.
CLASSROOM EXAMPLE 2
Applications
CLASSROOM EXAMPLE 3
(Home: copy Text Ex. 3 & 4 page 293)
Two boats leave a harbor at the same time, traveling on courses that make an angle of
82°20′ between them. When the slower boat has traveled 62.5 km, the faster one has
traveled 79.4 km. At that time, what is the distance between the boats?
■Formulas for Area of a Triangle (Sec. 3.1, 3.2)
1) Standard Formula:
2)
3)
Area of an Oblique Triangle (SAS)
In any triangle ABC, the area
Sec. 3.1 page 286
is given by the following formulas.
1
 bc sin A,
2

1
ab sin C ,
2
Finding the Area of a Triangle (SAS)
Find the area of triangle DEF in the figure.
CLASSROOM EXAMPLE 4
1
 ac sin B
2
Heron’s Area Formula (SSS) Sec. 3.2 page 294
In a triangle has sides of lengths a, b, and c, with semiperimeter
the area
s
1
a  b  c ,
2
of the triangle is given by the following formula.
 s  s  a  s  b  s  c 
CLASSROOM EXAMPLE 5
Using Heron’s Formula to Find an Area (SSS)
The distance “as the crow flies” from Chicago to St. Louis is 262 mi, from St. Louis to
New Orleans is 599 mi, and from New Orleans to Chicago is 834 mi. What is the area of the triangular region
having these three cities as vertices?
Answer: Area = 43 sq ft
Finding the Area of a Triangle (ASA)
Find the area of triangle ABC if B  5810, a = 32.5 cm, and C  7330.
CLASSROOM EXAMPLE 6
Answer: Area = 576 sq cm
Heron’s Formula for the Area of a Triangle
Heron’s Area Formula (SSS)
If a triangle has sides of lengths a, b, and c, with semiperimeter
s
then the area
1
a  b  c,
2
of the triangle is given by the following formula.
 s  s  a  s  b  s  c 
CLASSROOM EXAMPLE 7
Using Heron’s Formula to Find an Area (SSS)
The distance “as the crow flies” from Chicago to St. Louis is 262 mi, from St. Louis to
New Orleans is 599 mi, and from New Orleans to Chicago is 834 mi. What is the area of the triangular region
having these three cities as vertices?
Home: Copy Ex. 1, 2, 3, 4 (Sec. 3.2). Review for Exam 4.