7.7 Law of Cosines
• Use the Law of Cosines to solve triangles
and problems

Previously, we learned the Law of Sines, which as
some theorems can, it does have its limitations. To
use the Law of Sines we had to know the
measures of two angles and any side (AAS or ASA)
OR the measures of two sides and an opposite
angle (SSA).

If we don’t have either of these scenarios, but
instead we have the measures of two sides and
the included angle (SAS) or the measures of all the
sides and no angles (SSS), we must use the Law of
Cosines.
a
b
A

Law of Cosines:
In ABC,
a2 = b2 + c2 – 2bc cos A
b2 = a2 + c2 – 2ac cos B
c2 = a2 + b2 – 2ab cos C
c
b
C
a
B
Use the Law of Cosines since the measures of two sides
and the included angle are known.
Law of Cosines
Simplify.
Take the square root
of each side.
Use a calculator.
Answer:
Answer:
Law of Cosines
Simplify.
Subtract 754 from each side.
Divide each side by –270.
Solve for L.
Use a calculator.
Answer:
Answer:
Determine whether the Law of Sines or the Law of
Cosines should be used first to solve
Then
solve
Round angle measures to the nearest
degree and side measures to the nearest tenth.
Since we know the measures of two sides and the
included angle, use the Law of Cosines.
Law of Cosines
Take the square root
of each side.
Use a calculator.
Next, we can find
If we decide to find
we can use either the Law of Sines or the Law of Cosines
to find this value. In this case, we will use the Law of
Sines.
Law of Sines
Cross products
Divide each side by 46.9.
Take the inverse of each
side.
Use a calculator.
Use the Angle Sum Theorem to find
Angle Sum Theorem
Subtract 168 from each
side.
Answer:
Determine whether the Law of Sines or the Law of
Cosines should be used first to solve
Then
solve
Round angle measures to the nearest
degree and side measures to the nearest tenth.
Answer:
AIRCRAFT From the diagram of the plane shown,
determine the approximate exterior perimeter of each
wing. Round to the nearest tenth meter.
Since
is an isosceles triangle,
Use the Law of Sines to find KJ.
Law of Sines
Cross products
Divide each side by sin
Simplify.
.
Use the Law of Sines to find
.
Law of Sines
Cross products
Divide each side by 9.
Solve for H.
Use a calculator.
Use the Angle Sum Theorem to find
Angle Sum Theorem
Subtract 95 from each
side.
Use the Law of Sines to find HK.
Law of Sines
Cross products
Divide each side by sin
Use a calculator.
The perimeter of the wing is equal to
Answer: The perimeter is about
about 67.1 meters.
or
The rear side window of a station wagon has the
shape shown in the figure. Find the perimeter of the
window if the length of DB is 31 inches.
Answer: about 93.5 in.

Pre-AP Geometry:
Pg. 388 #12 – 38 evens, 40 & 42

Geometry:
Pg. 388 #11 & 12, 15 & 16,
19 & 20, 22 & 23,
27 – 30, 42