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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
OBJECTIVES
The Law of Cosines
Learn the statement and the derivation of the Law of Cosines.Learn to use the Law of Cosines to solve SAS triangles.Learn to use the Law of Cosines to solve SSS triangles. Learn to state and derive Heron’s formula for the area of a triangle.
SECTION 7.2
1
2
3
4
Slide 7.2 - 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
LAW OF COSINES
The following diagrams illustrate the Law of Cosines.
Slide 7.2 - 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
LAW OF COSINES
a2 b2 c2 2bccosA,
b2 c2 a2 2cacosB,
c2 a2 b2 2abcosC.
Let A, B, and C denote the measures of the angles of a triangle ABC, with opposite sides of lengths a, b, and c, respectively. Then
In words, the square of any side of a triangle is equal to the sum of the squares of the length of the other two sides, less twice the product of the lengths of the other sides and the cosine of their included angle.
Slide 7.2 - 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
SOLVING SAS TRIANGLES
Step 1: Use the appropriate form of the Law of Cosines to find the side opposite the given angle.
Step 2: Use the Law of Sines to find the angle opposite the shorter of the two given sides. Note that this angle is always an acute angle.
Step 3: Use the angle sum formula to find the third angle.
Step 4: Write the solution.
Slide 7.2 - 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1 Solving an SAS Triangle
Solve triangle ABC with b = 16 meters, c = 12 meters, and A = 50º. Round each answer to the nearest tenths. SolutionStep 1 Find a, the length of the side opposite
angle A. a2 b2 c2 2bccosA
a b2 c2 2bccosA
a 12.376 meters
a 16 2 12 2 2 16 12 cos 50º
Slide 7.2 - 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solution continued
sinC
c
sinA
a
EXAMPLE 1 Solving an SAS Triangle
Step 2 Find C, the measure of the angle opposite the shorter of the two given sides.
sinC csinA
a
C sin 1 csinA
a
C sin 1 12sin 50º
12.376
48º
Slide 7.2 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solution continued
B180º A C
EXAMPLE 1 Solving an SAS Triangle
Step 3 Find the third angle measure, B.
B180º 50º 48º
B82ºStep 4 The solution of triangle ABC is:
c = 12 metersC ≈ 48º
b = 16 metersB ≈ 82º
a ≈ 12.4 metersA = 50º
Slide 7.2 - 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2 Using the Law of Cosines
Suppose that a Boeing 747 is flying over Disney World headed due south at 552 miles per hour. Twenty minutes later, an F-16 passes over Disney World with a bearing of N 37º E at a speed of 1250 miles per hour. Find the distance between the two planes 3 hours after the F-16 passes over Disney World. Round the answer to the nearest tenth.
Slide 7.2 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2 Using the Law of Cosines
1
3
SolutionSuppose the F-16 has been traveling for t hours after passing over Disney World. Then, because the Boeing 747 had a head start of 20 minutes
hour, the Boeing 747 has
been traveling
hours due south.
t 1
3
The distance between the two planes is d.
Slide 7.2 - 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2 Using the Law of Cosines
d 2 1250t 2 552 t 1
3
2
2 1250t 552 t 1
3
cos143º
Solution continuedUsing the Law of Cosines in triangle FDB, we have
Substitute t = 3.
d 2 28, 469,270.04
d 5335.7 miles
Slide 7.2 - 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
SOLVING SSS TRIANGLES
Step 1: Use the Law of Cosines to find the side angle opposite the longest side.
Step 2: Use the Law of Sines to find either of the two remaining acute angles.
Step 3: Use the angle sum formula to find the third angle.
Step 4: Write the solution.
Slide 7.2 - 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3 Solving an SSS Triangle
Solve triangle ABC with a = 8, b = 5, and c = 7. Round each answer to the nearest tenths.
SolutionStep 1 Find A, the angle opposite the largest
side. a2 b2 c2 2bccosA
cosAb2 c2 a2
2bc
Acos 1 0.1429
52 72 82
257cosA0.1429
A81.8º
Slide 7.2 - 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solution continued
sinB
b
sinA
a
EXAMPLE 3 Solving an SSS Triangle
Step 2 Find B, using the Law of Sines.
sinBbsinA
a
Bsin 1 bsinA
a
C sin 1 5sin 81.8º
8
38.2º
Slide 7.2 - 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solution continued
B180º A B
Step 3 Find C by using the angle sum formula.
B180º 81.8º 38.2º
B60ºStep 4 Write the solution.
c = 7C ≈ 60º
b = 5B ≈ 38.2º
a = 8A ≈ 81.8º
EXAMPLE 3 Solving an SSS Triangle
Slide 7.2 - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4 Solving an SSS Triangle
Solve triangle ABC with a = 2 meters, b = 9 meters, and c = 5 meters. Round each answer to the nearest tenths. SolutionStep 1 Find B, the angle opposite the longest
side. b2 c2 a2 2cacosB
cosBc2 a2 b2
2ca
52 22 92
252cosB 2.6
Range of the cosine function is [–1, 1], there is no angle B, the triangle cannot exist.
Slide 7.2 - 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
HERON’S FORMULA FOR SSS TRIANGLES
The area K of a triangle with sides of lengths a, b, and c is given by
where is the semiperimeter.
K s s a s b s c ,
s 1
2a b c
Slide 7.2 - 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 6 Using Heron’s Formula
A triangular swimming pool has side lengths 23 feet, 17 feet, and 26 feet. How many gallons of water will fill the pool to a depth of 5 feet? Round answer to the nearest whole number.
SolutionTo calculate the volume of water in the swimming pool, we first calculate the area of the triangular surface.
We have a = 23, b = 17, and c = 26.
s 1
2a b c 1
22317 26 33
Slide 7.2 - 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solution continued
K s s a s b s c
192.2498 5
By Heron’s formula, the area K of the triangula surface is
EXAMPLE 6 Using Heron’s Formula
K 33 33 23 33 17 33 26 K 192.2498 square feet
Volume of water in pool = surface area depth
961.25 cubic feet