NAME
DATE
12-5
PERIOD
Study Guide and Intervention
Law of Cosines
Use Law of Cosines to Solve Triangles
Law of Cosines
C
a
b
A
Let ABC be any triangle with a, b, and c representing the measures of the sides,
and opposite angles with measures A, B, and C, respectively. Then the following
equations are true.
c
a2 = b2 + c2 - 2bc cos A
B
b2 = a2 + c2 - 2ac cos B
c2 = a2 + b2 - 2ab cos C
You can use the Law of Cosines to solve any triangle if you know the measures of two sides
and the included angle (SAS case), or the measures of three sides (SSS case).
Example
Solve ABC.
A
You are given the measures of two sides and the included angle.
Begin by using the Law of Cosines to determine c.
c2 = a2 + b2 - 2ab cos C
c2 = 282 + 152 - 2(28)(15)cos 82°
c2 ≈ 892.09
c ≈ 29.9
Next you can use the Law of Sines to find the measure of angle A.
15
C
c
82°
28
B
sin A
sin 82°
−
≈−
28
29.9
sin A ≈ 0.9273
A ≈ 68°
The measure of B is about 180° - (82° + 68°) or about 30°.
Exercises
Solve each triangle. Round side lengths to the nearest tenth and angle measures
to the nearest degree.
1. a = 14, c = 20, B = 38°
2. A = 60°, c = 17, b = 12
3. a = 4, b = 6, c = 3
4. A = 103°, b = 31, c = 52
5. a = 15, b = 26, C = 132°
6. a = 31, b = 52, c = 43
Chapter 12
29
Lesson 12-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
sin A
sin C
−
a =−
c
Glencoe Algebra 2
NAME
DATE
12-5
PERIOD
Study Guide and Intervention
(continued)
Law of Cosines
Choose a Method to Solve Triangles
Solving an
Oblique Triangle
Given
Begin by Using
two angles and any side
two sides and an angle opposite one of them
two sides and their included angle
three sides
Law
Law
Law
Law
Example
Determine whether ABC should be
solved by beginning with the Law of Sines or Law of
Cosines. Then solve the triangle.
of
of
of
of
Sines
Sines
Cosines
Cosines
a
B
8
You are given the measures of two sides and their included
angle, so use the Law of Cosines.
Law of Cosines
a2 = b2 + c2 - 2bc cos A
2
2
2
a = 20 + 8 - 2(20)(8) cos 34°
b = 20, c = 8, A = 34°
2
a ≈ 198.71
Use a calculator to simplify.
a ≈ 14.1
Use a calculator to simplify.
C
20
34°
A
Use the Law of Sines to find C.
sin C
sin A
−
Law of Sines
c =−
a
8 sin 34°
sin C ≈ −
c = 8, A = 34°, a ≈ 14.1
14.1
Exercises
Determine whether each triangle should be solved by beginning with the Law of
Sines or Law of Cosines. Then solve the triangle.
1. B
2. A
18
8
A
4
25°
b
4. A = 58°, a = 12, b = 8
Chapter 12
3.
C
C
B
22
16
128°
9
5. a = 28, b = 35, c = 20
30
B
A
20
C
6. A = 82°, B = 44°, b = 11
Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
C ≈ 18°
Use the sin-1 function.
The measure of angle B is approximately 180° - (34° + 18°) or about 128°.