23-3 and 23-4: The Law of Cosines
Objectives:
1. To derive and use
the Laws of Cosines
Assignment:
• P. 329-330: 9-15
• Challenge Problems
Objective 1
You will be able to derive and use
the Law of Cosines
Example 1
Find the length of 𝐴𝐶 in acute triangle 𝐴𝐵𝐶.
Law of Sines
As the previous example
demonstrates, you cannot
always use the Law of
Sines for every triangle.
You need either two sides
and an angle or two
angles and a side in the
following configurations:
ASA
AAS
SSA
Law of Cosines
To derive the Law of
Cosines, we need an
interesting diagram
like the one shown.
Click the picture for a
Geometer’s
Sketchpad
Demonstration of this
useful equation.
Law of Cosines
If 𝐴𝐵𝐶 has sides of
length a, b, and c as
shown, then
c 2  a 2  b 2  2ab  cos C
b 2  a 2  c 2  2ac  cos B
a 2  b 2  c 2  2bc  cos A
Example 2
Simplify the equation below for C = 90°.
c 2  a 2  b 2  2ab  cos C
General Pythagorean Theorem
When 𝐶 is acute, cos 𝐶 is positive:
Acute:
c2  a 2  b2
Need to subtract
something to = c2
c 2  a 2  b 2  2ab  cos C
General Pythagorean Theorem
When 𝐶 is obtuse, cos 𝐶 is negative:
Obtuse:
c2  a 2  b2
Need to add
something to = c2
c 2  a 2  b 2  2ab  cos C
This becomes minus a
negative, so you are
actually adding something
Example 3
Solve the equation below for C.
c 2  a 2  b 2  2ab  cos C
Example 1 (Revisited)
Find the length of 𝐴𝐶 in acute triangle 𝐴𝐵𝐶.
Example 4
Solve Δ𝐴𝐵𝐶.
Pro Tip: SSS
When using the Law of
Cosines to find a missing
angle (SSS), it’s a good
idea to find the angle
opposite the longest side
first. This is just in case
the angle turns out to be
obtuse. Regardless of
what type of angle this
turns out to be, use the
Law of Sines and the
Triangle Sum Theorem to
find the other two angles.
1: Use the Law of Cosines
2: Use the Law of Sines
3: Use the Triangle Sum
Example 5
Find the indicated measure.
1. x and y
2.  and 
Summary
Law of
Sines
Law of
Cosines
• ASA
• AAS
• SSA
• SAS
• SSS
Given three pieces of
any triangle, you
can use the Law of
Sines or the Law of
Cosines to
completely solve
the triangle.
Example 6
Given that 𝑚∠𝐵𝐴𝐷 = 101°, find the lengths
of 𝐴𝐵, 𝐴𝐵, and 𝐴𝐶.
C
11.11 ft
B
7.44 ft
A
D
23-3 and 23-4: The Law of Cosines
Objectives:
1. To derive and use
the Laws of Cosines
Assignment:
• P. 329-330: 9-15
• Challenge Problems