Trigonometry

1
Trigonometry

• Working with Oblique Triangles
• Law of Sines

2
Introduction

• Given a right triangle, you should feel
  comfortable using the three basic trig functions
  to determine additional information about the
  triangle.

3
Introduction

• Depending upon the information given, you could
  either determine the size of an angle...

opposite
hypotenuse
4
Introduction

• or determine the length of a side.

hypotenuse
adjacent
5
Introduction

• But what happens when you must work with an
  oblique triangle (one without a right angle)?
• Sometimes you can split-up an oblique triangle
  into right triangles...

6
Introduction

• but often it is better to use a more appropriate
  tool for the job.
• One of those tools for working with obtuse
  triangles is called the Law of Sines.

7
Law of Sines

• The Law of Sines states that in any triangle, the
  sides are proportional to the sines of the
  opposite angles.

C
a
b
A
B
c
8
Law of Sines

• Side a is proportional to the sine of angle
  A...

C
a
b
A
B
c
9
Law of Sines

• side b is proportional to the sine of angle
  B...

C
a
b
A
B
c
10
Law of Sines

• and side c is proportional to the sine of angle
  C.

C
a
b
A
B
c
11
Law of Sines

• There are two kinds of situations where you can
  use the Law of Sines.

• Situation 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.
12
Law of Sines

• There are two kinds of situations where you can
  use the Law of Sines.

• Situation 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.
13
Law of Sines

• There are two kinds of situations where you can
  use the Law of Sines

• Situation 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.
14
Law of Sines

• There are two kinds of situations where you can
  use the Law of Sines

• Situation 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.
15
Example 1
16
Law of SinesExample 1

• Given the diagram below, determine the length of
  side x.

17
Law of SinesExample 1

• What do we know about this problem?
• First of all, it is an oblique triangle.
• Second, we note that two angles are known, and
  one of the sides opposite.

18
Law of SinesExample 1

• Thats enough info to verify that using the Law
  of Sines will allow us to determine the length of
  x.

19
Law of SinesExample 1

• To solve, set up a proportion. Remember that the
  sides are proportional to the sines of the
  opposite angles.

20
Law of SinesExample 1

• Start by pairing the 63 angle and the 5.45 side
  together since they are opposite one another.

21
Law of SinesExample 1

• The unknown side x is opposite the 47 angle.
  Pair these up to complete the proportion.

22
Law of SinesExample 1

• Solve the proportion by cross-multiplying.

Multiply on this diagonal first. 5.45 x sin47
3.99
23
Law of SinesExample 1

• Solve the proportion by cross-multiplying.

Next, divide 3.99 by sin63 3.99 sin63
4.47 (this is the length of x)
24
Law of SinesExample 1

• By using the Law of Sines, we know the length of
  side x is 4.47 inches.

4.47
25
Example 2
26
Law of SinesExample 2

• Given the diagram below, determine the length of
  side x.

27
Law of SinesExample 2

• Before you jump in, be sure you know what you are
  dealing with.
• You are working with an oblique triangle...
• and you know two angles and a side opposite one
  of those angles.

28
Law of SinesExample 2

• That means using the Law of Sines will allow you
  to solve for x.

29
Law of SinesExample 2

• Set-up a proportion, starting with the 65.85
  mm side and the 85 angle since they are opposite
  one another.

30
Law of SinesExample 2

• Then complete the proportion by making another
  ratio using side x and the 42 angle.

31
Law of SinesExample 2

• Solve the proportion.

Multiply on this diagonal first. 65.85 x sin42
44.1
32
Law of SinesExample 2

• Solve the proportion.

Next, divide 44.1 by sin85 44.1 sin85
44.2 (this is the length of x)
33
Law of SinesExample 2

• Using the Law of Sines on this problem gives you
  an answer of 44.2 mm.

42
85
44.2 mm
65.85 mm
34
Example 3
35
Law of SinesExample 3

• Try this one on your own.
• Set-up a proportion and solve for x.
• Then click to see the answer.

9.25 cm
88
x
57
36
Law of SinesExample 3

• How did it turn out?

x 11.02 cm
9.25 cm
88
x
57
37
Law of Sines

• Recall that the other scenario where you can use
  the Law of Sines is when you know the lengths of
  two sides and the size of an angle opposite on of
  those sides.

38
Example 4
39
Law of SinesExample 4

• Given the diagram below, determine the size of
  angle A.

A
40
Law of SinesExample 4

• Once again, set-up a proportion.
• Start by pairing-up the 70 mm side and the 42
  angle.

A
41
Law of SinesExample 4

• Complete the proportion by putting the 85.5 mm
  side and angle A together.

A
42
Law of SinesExample 4

• This proportion will be a little more difficult
  to solve. The steps are shown below

Cross-multiply on this diagonal...
43
Law of SinesExample 4

• This proportion will be a little more difficult
  to solve. The steps are shown below

then multiply on this diagonal...
44
Law of SinesExample 4

• This proportion will be a little more difficult
  to solve. The steps are shown below

Divide both sides by 70.
45
Law of SinesExample 4

• This proportion will be a little more difficult
  to solve. The steps are shown below

Evaluate the left side of the equation.
46
Law of SinesExample 4

• This proportion will be a little more difficult
  to solve. The steps are shown below

Type 0.8173 into your calculator, press the 2nd
function key, then press the sin key.
47
Law of SinesExample 4

• So the size of angle A is 54.8.

54.8
48
Example 5
49
Law of SinesExample 5

• Given the diagram below, determine the size of
  angle A.

50
Law of SinesExample 5

• Once again, set-up a proportion.
• Start by pairing-up the 4.2 side and the 55
  angle.

51
Law of SinesExample 5

• Complete the proportion by putting the 4.9 side
  and angle A together.

52
Law of SinesExample 5

• Follow the steps shown to solve the proportion

Cross-multiply on this diagonal...
53
Law of SinesExample 5

• Follow the steps shown to solve the proportion

then multiply on this diagonal...
54
Law of SinesExample 5

• Follow the steps shown to solve the proportion

Divide both sides by 4.2.
55
Law of SinesExample 5

• Follow the steps shown to solve the proportion

Evaluate the left side of the equation.
56
Law of SinesExample 5

• Follow the steps shown to solve the proportion

Type 0.9557 into your calculator, press the 2nd
function key, then press the sin key.
57
Law of SinesExample 5

• You have just determined that angle A is 72.9.

4.2
72.9
55
4.9
58
Example 6
59
Law of SinesExample 6

• Try this one on your own.
• Set-up a proportion and solve for angle A.
• Then click to see the answer.

10.8
A
12.25
60
60
Law of Sines Example 6

• The set-up and answer are shown below

x 79.2
10.8
A
12.25
60
61
End of Lesson