13-6 The Law of Cosines
Objectives
Use the Law of Cosines to solve triangles.
Apply Area Formula to triangles.
Holt Algebra 2
13-6 The Law of Cosines
Notes #1-4
Use the given measurements to solve ∆ABC (nearest tenth).
1. a = 18, b = 40, m
C = 82.5°
2. x = 18; y = 10; z = 9
3. Two model planes take off from the same spot. The first plane
travels 300 ft due west before landing and the second plane
travels 170 ft southeast before landing. To the nearest foot,
how far apart are the planes when they land?
4. An artist needs to know the area of a triangular piece of
stained glass with sides measuring 9 cm, 7 cm, and 5 cm.
What is the area to the nearest square centimeter?
Holt Algebra 2
13-6 The Law of Cosines
In the previous lesson, you learned to solve triangles by using the
Law of Sines. However, the Law of Sines cannot be used to solve
triangles for which side-angle-side (SAS) or side-side-side (SSS)
information is given. Instead, you must use the Law of Cosines.
Holt Algebra 2
13-6 The Law of Cosines
Example 1: Using the Law of Cosines
Use the given measurements to solve ∆ABC.
Round to the nearest tenth.
a = 8, b = 5, mC = 32.2°
Step 1 Find the length of the third side.
c2 = a2 + b2 – 2ab cos C
Law of Cosines
c2 = 82 + 52 – 2(8)(5) cos 32.2°
Substitute.
c2 ≈ 21.3
c ≈ 4.6
Holt Algebra 2
Use a calculator to
simplify.
Solve for the positive
value of c.
13-6 The Law of Cosines
Example 1 Continued
Step 2 Find the measure of the smaller angle, B.
Law of Sines
Substitute.
Solve for m
Step 3 Find the third angle measure.
mA  112.4°
Holt Algebra 2
B.
13-6 The Law of Cosines
Example 2: Using the Law of Cosines
Use the given measurements to solve ∆ABC.
Round to the nearest tenth.
a = 8, b = 9, c = 7
Step 1 Find the measure of the largest angle,
B.
b2 = a2 + c2 – 2ac cos B
Law of cosines
92 = 82 + 72 – 2(8)(7) cos B
cos B = 0.2857
Substitute.
Solve for cos B.
m
Solve for m B.
Holt Algebra 2
B = Cos-1 (0.2857) ≈ 73.4°
13-6 The Law of Cosines
Example 2 Continued
Use the given measurements to solve ∆ABC (nearest tenth).
Step 2 Find another angle measure
72 = 82 + 92 – 2(8)(9) cos C
Substitute Law of Cosines
cos C = 0.6667
Solve for cos C.
m
Solve for m C.
C = Cos-1 (0.6667) ≈ 48.2°
Step 3 Find the third angle measure.
m
A  58.4°
Holt Algebra 2
13-6 The Law of Cosines
Remember!
The largest angle of a triangle is the angle
opposite the longest side.
When using the LAW of COSINES, find the largest angle first.
When using the LAW of SINES, find the largest angle last
(using the triangle sum formula)
Holt Algebra 2
13-6 The Law of Cosines
Example 3
The surface of a hotel swimming pool is shaped like a
triangle with sides measuring 50 m, 28 m, and 30 m.
What is the area of the pool’s surface to the nearest
square meter?
Find the measure of the largest angle, A.
502 = 302 + 282 – 2(30)(28) cos A
m
A ≈ 119.0°
Law of Cosines
Solve for m

Holt Algebra 2
A.
13-6 The Law of Cosines
Notes
Use the given measurements to solve ∆ABC.
Round to the nearest tenth.
1. a = 18, b = 40, m C = 82.5°
c ≈ 41.7; m A ≈ 25.4°; m B ≈ 72.1°
2. x = 18; y = 10; z = 9
m Z ≈ 142.6°; m Y ≈ 19.7°; m Z ≈ 17.7°
Holt Algebra 2
13-6 The Law of Cosines
Notes #3-4
3. Two model planes take off from the same spot.
The first plane travels 300 ft due west before
landing and the second plane travels 170 ft
southeast before landing. To the nearest foot,
how far apart are the planes when they land?
437 ft
4. An artist needs to know the area of a
triangular piece of stained glass with sides
measuring 9 cm, 7 cm, and 5 cm. What is the
area to the nearest square centimeter?
17 cm2
Holt Algebra 2