Download Heron`s Formula can be used to find the area of a

Unit 6 – Law of Sines &
Precalculus
Cosines
Name
Date
Friday April 25, 2014
Notes 6.2 Law of Cosines & Heron’s Area Formula
So far this year, we’ve been working exclusively with right triangles. Accordingly, we could use the Pythagorean Theorem and/or
Right Triangle Trigonometry with SOH CAH TOA to find the missing sides and angles of these right triangles. Now it is time to
learn the methods that will help us find these missing measures when the triangle is not a right triangle. The Law of Sines and the
Law of Cosines allow us to find the missing measures of oblique triangles. (Oblique triangles are triangles without a right angle.)
The Law of Sines can be used if you are given:
The Law of Cosines can be used if you are given:
 Two angles and a non-included side (AAS)
 Two angles and an included side (ASA)
 Two sides and a non-included angle (SSA)*
(SSA is referred to as the ambiguous case since it could
have no triangle, one triangle, or two triangles.)


Three sides (SSS)
Two sides and an included angle (SAS)
c 2  a 2  b 2  2ab cos C
or
a  b  c  2bc cos A
2
a
b
c


sin A sin B sin C
2
2
or
b 2  a 2  c 2  2ac cos B
Important Note: You MUST use the Law of Cosines to find the largest angle of
the triangle BEFORE switching to Law of Sines. Failing to follow this rule could
result in creating an unnecessary and incorrect ambiguous case accidentally. 
Both laws require that you to know how to label a triangle properly.
C
a
b
A
Name angles with capital/upper case letters.
Name sides directly across from the angle with the same variable using lower case letters.
B
c
1. Solve the triangle. (This means to find all the missing angle measurements and side lengths.)
a = 3, b = 5, c = 7
C
Start with
Since given info is SSS (SideSideSide),
start with law of cosines
Law of Cosines always has 1 triangle.
# of ∆(s):
3
5
B
7
Note the relationship between the initial squared side and the indicated angle.
72  32  52  2(3)(5) cos C
49  9  25  30 cos C
but does not become 49  4cos C 
49  34  30 cosC
15  30cosC
120  C
1
2
=  B = 38.20
=  C = 1200
c 2  a 2  b 2  2ab cos C
1
 cos C
2
 1
cos 1     C
 2
Law of Cosines
no triangle
=  A = 21.80
Since “c” is the longest side,  C will be the
Abiggest angle and we find this one first.

Law of Sines
=a=
3
=b=
5
When evaluating with the Law of Cosines, a common
= cERROR
=
7is to
combine the coefficient of cos  with the squared terms (the terms shown
in red and blue) Be sure to use the correct order of operations when
simplifying the expression.
b 2  a 2  c 2  2ac cos B
a 2  b 2  c 2  2bc cos A
 b2  a 2  c 2 
B  cos1 

 2ac 
 52  32  7 2 
B  cos 1 

  2  3 7   


B  38.2
 a 2  b2  c 2 
A  cos1 

 2bc 
 32  52  7 2 
A  cos 1 

  2  5  7   



A  21.8
Solve each triangle. (This means to find all the missing angle measurements and side lengths.)
Round any side lengths to the nearest hundredth and round angle measurements to the nearest tenth of a degree.
Appropriate work must support your answer to receive full credit for correct responses.
2.  B = 98.1, a = 17, c = 21
Start with
# of ∆(s):
Law of Sines
Law of Cosines
no triangle
1
2
= A =
98.10
= B =
= C =
=a=
17
=b=
=c=
3.
a = 43, b = 52, c = 39
Start with
# of ∆(s):
21
Law of Sines
Law of Cosines
no triangle
1
2
= A =
= B =
= C =
=a=
43
=b=
52
=c=
39
Caution: Don’t round until the final step of getting an answer!!!
4.
To approximate the length of a lake, a surveyor starts at one end of the lake and walks 245 yards. He then turns
110º and walks 270 yards until he arrives at the other end of the lake. Approximately how long is the lake?
Round to the nearest yard.
Heron’s Formula can be used to find the area of a triangle when all three sides are known
Area =
s ( s  a )( s  b)( s  c) , where s 
abc
and a, b, and c are lengths of the sides.
2
s is known the “semiperimeter” which can help you in remembering this formula as half of the perimeter.
5.
Find the area of a triangle with sides 6 cm, 9 cm, and 13 cm. Round the area to the nearest square centimeter.
6.
Find the area of the Bermuda Triangle if the sides of the triangle have the approximate lengths 850 miles, 925
miles, and 1300 miles. Round the area to the nearest square mile.