Trigonometry Working with Oblique Triangles: Law of Cosines

1
Trigonometry

• Working with Oblique Triangles
• Law of Cosines

2
Introduction

• If you have viewed the lesson on the Law of
  Sines, you know that the formula is useful for
  solving dimensions or angles found in oblique
  triangles.
• Oblique triangles are non-right triangles such as
  the ones shown below

3
Law of SinesReview of the Law of Sines

• Recall the conditions under which the Law of
  Sines is used

• Application 1 When you know two angles, the
  length of a side opposite, and you want to
  determine the length of another side.

This known side is opposite one of the given
angles.
4
Law of SinesReview of the Law of Sines

• Recall the conditions under which the Law of
  Sines is used

• Application 1 When you know two angles, the
  length of a side opposite, and you want to
  determine the length of another side.

This side is opposite the other given angle.
5
Law of SinesReview of the Law of Sines

• Recall the conditions under which the Law of
  Sines is used

• Application 2 You know the length of two sides,
  the size of an angle opposite, and you want to
  determine the size of another angle.

This angle is opposite one of the given sides.
6
Law of SinesReview of the Law of Sines

• Recall the conditions under which the Law of
  Sines is used

• Application 2 You know the length of two sides,
  the size of an angle opposite, and you want to
  determine the size of another angle.

This unknown angle is opposite the other side.
7
IntroductionThe Law of Cosines formula.

• There is another formula for oblique triangles
  called the Law of Cosines.

C
b
a
A
B
c
8
IntroductionThe Law of Cosines formula.

• There is another formula for oblique triangles
  called the Law of Cosines.

In the diagram below, the upper case letters
represent angles...
C
b
a
A
B
c
9
IntroductionThe Law of Cosines formula.

• There is another formula for oblique triangles
  called the Law of Cosines.

...and the lower case letters represent the
length of sides opposite those angles.
C
b
a
A
B
c
10
IntroductionThe Law of Cosines formula.

• There is another formula for oblique triangles
  called the Law of Cosines.

Dont worry about how it works yet!
C
b
a
A
B
c
11
Law of Cosines
Application 1
12
Law of CosinesApplication 1

• With the Law of Cosines you can do the following

• Application 1 You can determine the length of a
  side if you know two sides and the included angle.

This is called the included angle because it is
the angle formed by the two known sides (shown in
red).
13
Law of Cosines
EXAMPLE 1
14
Law of CosinesExample 1

• In the diagram below, you can use the Law of
  Cosines to determine the length of side x.

and the included angle.
70
25 mm
19 mm
Therefore, we can compute the length of this side
which is opposite the included angle.
x
15
Law of CosinesExample 1

• With the given information, we will plug-in the
  numbers into the formula.

70
25 mm
19 mm
x
16
Law of CosinesExample 1

• First, lets take a tour of the formula.
• Watch as we update the formula with numbers from
  our example.

a is reserved for the unknown side.
Watch the formula get updated here
70
25 mm
19 mm
x
17
Law of CosinesExample 1
Watch the formula get updated here
70
25 mm
19 mm
x
18
Law of CosinesExample 1
Watch the formula get updated here
70
25 mm
19 mm
x
19
Law of CosinesExample 1
A is the size of the included angle.
Watch the formula get updated here
70
25 mm
19 mm
x
20
Law of CosinesExample 1

• Now we have to solve the equation for x

21
Law of CosinesExample 1

• Now we have to solve the equation for x

22
Law of CosinesExample 1

• Now we have to solve the equation for x

To finish, take the square root of both sides of
the equation.
23
Law of CosinesExample 1

• Now we know the length of side x is 25.7 mm.

70
25 mm
19 mm
25.7 mm
24
Law of Cosines
EXAMPLE 2
25
Law of CosinesExample 2

• Lets try another problem.
• Solve for dimension x.

26
Law of CosinesExample 2

• We will be able to use the Law of Cosines to
  determine the length of x for the following
  reasons...

Therefore, we can compute the length of this side
which is opposite the included angle.
and the included angle.
27
Law of CosinesExample 2

• First, write down the Law of Cosines formula

a is reserved for the unknown side.
Watch the formula get updated here
28
Law of CosinesExample 2
Watch the formula get updated here
29
Law of CosinesExample 2
Watch the formula get updated here
30
Law of CosinesExample 2
A is the size of the included angle.
Watch the formula get updated here
31
Law of CosinesExample 2

• Now we have to solve the equation for x

32
Law of CosinesExample 2

• Now we have to solve the equation for x

33
Law of CosinesExample 2

• Now we have to solve the equation for x

To finish, take the square root of both sides of
the equation.
34
Law of CosinesExample 2

• Now we know the length of side x is 2.17 in.

35
Law of Cosines
EXAMPLE 3
36
Law of CosinesExample 3

• Solve for dimension x.
• Do this problem on your own. Then click to see
  the answer.

37
Law of CosinesExample 3

• Click to see the steps used to calculate the
  answer

38
Law of CosinesExample 3

• Click to see the steps used to calculate the
  answer

39
Law of CosinesExample 3

• Click to see the steps used to calculate the
  answer

To finish, take the square root of both sides of
the equation.
40
Law of CosinesExample 3

• The length of side x must be 8.9 inches.

41
Law of Cosines
Application 2
42
Law of CosinesApplication 2

• With the Law of Cosines you can also do the
  following

• Application 2 You can determine the size of an
  angle when the lengths of all three sides are
  known.

2
1.6
2.35
43
Law of CosinesApplication 2

• In order to determine the size of an angle, you
  must rearrange the Law of Cosines formula.

• The original formula

now looks like this
Lets put this formula to work on Example 4...
44
Law of Cosines
EXAMPLE 4
45
Law of CosinesExample 4

• Use the Law of Cosines to determine angle A.

A
39 mm
35 mm
22 mm
46
Law of CosinesExample 4

• We will use this arrangement of the Law of
  Cosines formula

A
39 mm
35 mm
22 mm
47
Law of CosinesExample 4

• Where do the numbers go into the formula?

A
39 mm
35 mm
22 mm
48
Law of CosinesExample 4
The side that is opposite the angle you are
trying to compute must go in a.
A
39 mm
35 mm
22 mm
49
Law of CosinesExample 4
b and c represent the other two sides of the
triangle.
A
39 mm
35 mm
22 mm
50
Law of CosinesExample 4
Use your calculator to evaluate the right side of
the equation.
A
39 mm
35 mm
22 mm
51
Law of CosinesExample 4
then press the cos key in order to determine the
size of the angle.
To finish, make sure your calculator display
shows 0.8286
press the 2nd function key...
A
39 mm
35 mm
22 mm
52
Law of CosinesExample 4

• Good work! We have used the Law of Cosines to
  calculate the size of angle A.

34
39 mm
35 mm
22 mm
53
Law of Cosines
EXAMPLE 5
54
Law of CosinesExample 5

• Use the Law of Cosines to determine angle A.

1.4
1.1
A
1.6
55
Law of CosinesExample 5

• Once again we will use this arrangement of the
  Law of Cosines

56
Law of CosinesExample 5
The side that is opposite the angle you are
trying to compute must go in a.

• Where do the numbers go into the formula?

57
Law of CosinesExample 5
b and c represent the other two sides of the
triangle.
58
Law of CosinesExample 5
Use your calculator to evaluate the right side of
the equation.
59
Law of CosinesExample 5
To finish, make sure your calculator display
shows 0.5142
then press the cos key in order to determine the
size of the angle.
press the 2nd function key...
60
Law of CosinesExample 5

• Excellent! We have used the Law of Cosines to
  calculate the size of angle A.

1.4
1.1
59.1
1.6
61
Law of Cosines
EXAMPLE 6
62
Law of CosinesExample 6

• Determine the size of angle A.
• Do this problem on your own. When you are done,
  click to see the answer worked out.

A
109 ft
115 ft
111 ft
63
Law of CosinesExample 6

• To solve for A, you must use this version of the
  Law of Cosines

A
109 ft
115 ft
111 ft
64
Law of CosinesExample 6

• Now plug-in the numbers from the diagram into the
  formula

A
109 ft
115 ft
111 ft
This is opposite angle A, so it must go in a.
65
Law of CosinesExample 6

• Now plug-in the numbers from the diagram into the
  formula

A
109 ft
115 ft
These two sides go in b and c.
111 ft
66
Law of CosinesExample 6

• Now solve the formula for angle A

A
109 ft
115 ft
111 ft
67
Law of CosinesExample 6

• You have calculated that angle A is 59.3.

59.3
109 ft
115 ft
111 ft
68
Law of CosinesReview
Review
69
Law of CosinesReview
In this lesson you have learned two applications
for the Law of Cosines formula

• Application 1
• When you know two sides and an included angle,
  you can compute the length of the side opposite
  the included angle.

70
Law of CosinesReview
In this lesson you have learned two applications
for the Law of Cosines formula

• Application 2
• When you know all three sides of a triangle, you
  can compute the size of any angle.

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End of Presentation