Trigonometry Chapter

1
Trigonometry Chapter

• Section 2 Right Triangle Trigonometry

2
Trigonometry Section 2

• Introduction

3
Applications
4
Trigonometry

• What can you use it for?

5
Naming the Sides of a Right Triangle
hypotenuse
leg
leg
6
Naming the Sides of a Right Triangle
A

• Names of the Sides
• hypotenuse
• opposite
• adjacent

Reference Angle
B
7
Naming the Sides of a Right Triangle
A

• Names of the Sides
• hypotenuse
• opposite
• adjacent

8
Naming the Sides of a Right Triangle

• Names of the Sides
• hypotenuse
• opposite
• adjacent

B
9
Naming the Sides of a Right Triangle

• Names of the Sides
• hypotenuse
• opposite
• adjacent

A
10
Practice Set 2

• Page 11

11
Trigonometry Section 2

• The Three Trigonometry Ratios

12
Ratios Review

• Ratios are a comparison between two numbers.
• Ratio
• 2-cycle engines gas to oil
• Metro area renters to home owners
• Metal shop contract work to custom work
• Format
• 4 to 5 45 4/5

13
Write a Ratio

• State the ratio of length to width of a picture
  frame with the following dimensions.

L14
W12
14
Trigonometry Ratios

• Use trig ratios to solve problems
• sine ratio
• cosine ratio
• tangent ratio

15
Sine Ratio

• Write the sine of angle B.

A
b
c
B
C
a
16
Sine Ratio

• Write the sine of angle A.

A
b
c
B
C
a
17
Sine Ratio

• Write the sine of angle A.

B
6
3
A
C
5.2
18
Sine Ratio

• Write the sine of angle Z.

Z
18
15.6
X
Y
9
19
Properties of Trig Ratios
10.8
5.4
7.2
4
3.6
2
30?
20
Practice Set 3

• Page 16

21
Trigonometry Section 2

• Sine ratios

22
Sine Ratios of Other Angles

• 0? lt angle lt 90?
• Each angle has a unique sine ratio.
• You can determine these ratios with 3 methods.

Graphical Approach
96.6 mm
25 mm
23
Sine Ratios of Other Angles

• 0? lt angle lt 90?
• Each angle has a unique sine ratio.
• You can determine these ratios with 3 methods.

Angle Sine Ratio
0? 0
15? 0.2588
30? 0.5
45? 0.7071
60? 0.8660
75? 0.9659
90? 1
Table
24
Sine Ratios of Other Angles

• 0? lt angle lt 90?
• Each angle has a unique sine ratio.
• You can determine these ratios with 3 methods.

1. Traditional Calculator
2. Direct Algebraic Logic (DAL)

Calculator
25
Determine the sine ratio calculator
Determine the sine ratio of 50?
sin
0.766044443
5
0

sin50?
0.766044443
sin
5
0
26
Sine Ratio with Calculator

• Try these with your calculator
• sin 1?
• sin 42?
• sin 80?
• sin 89.9?

0.01745
0.6691
0.9848
0.9999
27
Practice Set 4

• Pages 18 - 19

28
Trigonometry Section 2

• Importance of Sine Ratios

29
Using Sine Ratios
Angle Sine Ratio
0? 0
15? 0.2588
30? 0.5
45? 0.7071
60? 0.8660
75? 0.9659
90? 1

• Why is the sine ratio important?
• The sine ratio of 15? is _____.
• The sine ratio of 75? is _____.
• What angle has a sine ratio of 0.7071?
• What angle has a sine ratio of 0.8660?

30
Using Sine Ratios

• Determine the measure of Angle A

28
28
B
C
37
37
A
Table
0.7568
31
Sine Ratio ? Angle

• Determine the size of an angle that has a sine
  ratio of 0.7568
• sin A 0.7568
• arcsin(sin A)arcsin(0.7568)
• A arcsin(0.7568)
• A

32
Determine the Angle calculator
Determine the size of the angle which has a sine
ratio of 0.7568?
sin
7
.

49.18289932
2nd
5
6
8
sin A 0.7568
7
.
5
6
8
sin
49.18289932
2nd
33
Sine Ratio ? Angle

• Use your calculator to determine the size of each
  angle
• sin A 0.9903
• ?A 82?
• sin X 0.9397
• ?X 70?
• sin C 0.0523
• ?C 3?
• sin B 0.3090
• ?B 18?

34
Practice Set 5

• Page 22

35
Trigonometry Section 2

• Use Trig Ratios to Compute Angles and Lengths
• Sine Ratio

36
Determine Angles using Sine
Example 1 of 2

• Use the sine formula to determine the size of
  angle A.

41.8
15
A
39
37
Determine Angles using Sine
Example 2 of 2

• Use the sine function to determine the size of
  angle R.

6.4
4.1
7.6
R
38
Practice Set 6

• Page 25

39
Trigonometry Section 2

• Use Sine to Determine the Length of a Side (Part
  I)

40
Determine Lengths using Sine
Example 1 of 2

• Use the sine formula to determine the length of
  side x.

3.5
x
25?
x 1.48
41
Determine Lengths using Sine
Example 2 of 2

• Use the sine formula to determine the length of
  side x.

x
18
65?
x 16.3
42
Practice Set 7

• Page 28

43
Trigonometry Section 2

• Use Sine to Determine the Length of a Side (Part
  II)

44
Determine Lengths using Sine
Example 1 of 2

• Use the sine formula to determine the length of
  side x.

x
4
30?
x 8
45
Determine Lengths using Sine
Example 2 of 2

• Use the sine formula to determine the length of
  side x.

9
x
70?
x 9.6
46
Practice Set 8

• Page 31

47
Trigonometry Section 2

• Solving Lengths and Angles using the Sine formula.

48
Solving Triangles using Sine

• With the sine formula we can
• Determine an Angle
• Determine the Length of a Side
• Opposite Side
• Hypotenuse

49
Solving Triangles using Sine

• Solve for each missing dimension.

?A 27.5 deg. x 2.47 x 1.4
50
Practice Set 9

• Page 33

51
Trigonometry Section 2

• The cosine ratio

52
Trigonometry Ratios

• Use trig ratios to solve problems
• sine ratio
• cosine ratio
• tangent ratio

53
Cosine Formula

• The cosine of an angle

hyp
hyp
opp
A
adj
adj
54
Cosine Ratios

• Every angle from 0? to 90? has a unique cosine
  ratio.

55
Cosine Ratios

• Try these with your calculator
• cos 5?
• 0.9962
• cos 70?
• 0.3420

56
Cosine Ratios

• Try these on your calculator
• cos A 0.9397
• A 20?
• cos A 0.0872
• A 85?

57
Practice Set 10

• Page 36

58
Trigonometry Section 2

• Use cosine to compute the size of an angle.

59
Determine Angles using Cosine
Example 1 of 2

• Use the cosine formula to determine the size of
  angle A.

41.8
15
A
39
?A 21.1 deg.
60
Determine Angles using Cosine
Example 2 of 2

• Use the cosine function to determine the size of
  angle R.

6.4
4.1
7.6
R
?R 57.4 deg.
61
Practice Set 11

• Page 38

62
Trigonometry Section 2

• Use Cosine to Determine the Length of a Side

63
Determine Lengths using Cosine
Example 1 of 2

• Use the cosine formula to determine the length of
  side x.

x
30?
4
x 4.6
64
Determine Lengths using Cosine
Example 2 of 2

• Use the cosine formula to determine the length of
  side x.

x
18
65?
x 7.6
65
Practice Set 12

• Page 41

66
Trigonometry Section 2

• Solving Lengths and Angles using the Cosine
  formula.

67
Solving Triangles using Cosine

• With the cosine formula we can
• Determine an Angle
• Determine the Length of a Side
• Adjacent Side
• Hypotenuse

68
Solving Triangles using Sine

• Solve for each missing dimension.

?A 35.4 deg. x 7.6 x 0.54
69
Practice Set 13

• Page 43

70
Trigonometry Section 2

• The Tangent Ratio

71
Trigonometry Ratios

• Use trig ratios to solve problems
• sine ratio
• cosine ratio
• tangent ratio

72
Tangent Formula

• The tangent of an angle

hyp
opp
opp
A
adj
adj
73
Trigonometry Section 2

• Using the Tangent Function to Solve for Angles
  and Lengths of Sides

74
Solving Triangles using Tangent

• With the tangent formula we can
• Determine an Angle
• Determine the Length of a Side
• Opposite Side
• Adjacent Side

75
Solving Triangles using Tangent

• Solve for each missing dimension.

76
Practice Set 15

• Page 48