6.2
Law of Cosines
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What You Should Learn
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OK, I deleted the picture this time. I hope you
are having a good vacation and did not wait
until Sunday night to do this. 
Use the Law of Cosines to solve oblique
triangles (SSS or SAS).
Use the Law of Cosines to model and solve
real-life problems.
Use Heron’s Area Formula to find
areas of triangles.
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Introduction
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Introduction
Two cases remain in the list of conditions needed to solve
an oblique triangle—SSS and SAS.
To use the Law of Sines, you must know at least one side
and its opposite angle.
When you are given three sides (SSS), or two sides and
their included angle (SAS), none of the ratios in the Law of
Sines would be complete. In such cases you can use the
Law of Cosines.
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Introduction
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Example 1 – Three Sides of a Triangle—SSS
Find the three angles of the triangle shown in Figure 6.12.
Figure 6.12
Solution:
It is a good idea first to find the angle opposite the longest
side—side b in this case. Using the alternative form of the
Law of Cosines, you find that
Alternative form
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Example 1 – Solution
cont’d
Substitute for a, b, and c.
Simplify.
Because cos B is negative, you know that B is an obtuse
angle given by B  116.80.
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Example 1 – Solution
cont’d
At this point it is simpler to use the Law of Sines to
determine A.
You know that A must be acute, because B is obtuse, and
a triangle can have, at most, one obtuse angle.
So, A  22.08 and
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Heron’s Area Formula
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Heron’s Area Formula
The Law of Cosines can be used to establish the following
formula for the area of a triangle. This formula is called
Heron’s Area Formula after the Greek mathematician
Heron (ca. 100 B.C.). Memorizing this formula is optional.
You can usually use the formula for area from 6.1 instead if
you prefer.
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Example 5 – Using Heron’s Area Formula
Find the area of the triangle shown in Figure 6.16.
Figure 6.16
Solution:
Because
= 84
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Example 5 – Solution
cont’d
Heron’s Area Formula yields
Area
≈ 1131.89 square meters.
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Heron’s Area Formula
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