Introduction to Trigonometry

1
The Tangent Ratio
The Tangent using Angle
The Tangent Ratio in Action
The Tangent (The Adjacent side)
The Tangent (Finding Angle)
The Sine of an Angle
The Sine Ration In Action
The Sine ( Finding the Hypotenuse)
The Cosine of an Angle
Mixed Problems
2
Starter Questions
S3 Credit
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3
Angles Triangles
Learning Intention
Success Criteria

1. To identify the hypotenuse, opposite and adjacent
   sides in a right angled triangle.

1. Understand the terms hypotenuse, opposite and
adjacent in right angled triangle.
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2. Work out Tan Ratio.
4
Lets Investigate!
Trigonometry
S3 Credit
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5
Trigonometry means triangle and measurement.
We will be using right-angled triangles.

Opposite
hypotenuse
x
Adjacent
6
Mathemagic!
Opposite
hypotenuse
30
Adjacent
Opposite

0.6
Adjacent
7
Try another!
Opposite
hypotenuse
45
Adjacent
Opposite

1
Adjacent
8
For an angle of 30,
We write tan 30 0.6
9
S3 Credit
Tan 25 0.466
Tan 26 0.488
Tan 27 0.510
Tan 28 0.532
Tan 29 0.554
Tan 30 0.577
Tan 31 0.601
Tan 32 0.625
Tan 33 0.649
Tan 34 0.675
The ancient Greeks discovered this and repeated
this for all possible angles.
Accurate to 3 decimal places!
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On your calculator press
Tan
Followed by 30, and press

Notice that your calculator is incredibly
accurate!!
Accurate to 9 decimal places!
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How high is the tower?
60
12 m
13
Opposite
hypotenuse
60
12 m
Adjacent
14
Opp
Tan x
Adj
Opp
Tan 60
12
Opp
12 x Tan 60
Opp
12 x Tan 60
20.8m (1 d.p.)
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So the towers 20.8 m high!
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S3 Credit
Opp
Tan x
Adj
Opposite
x
Adjacent
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Example
Find the height h
S3 Credit
SOH CAH TOA
Opp
Hyp
Opp
h
Tan x
Adj
65
h
Tan 65
8m
8
Adj
h
8 x Tan 65
h
8 x Tan 65
17.2m (1 d.p.)
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Class Group Identifying the Tan Ratio Ex 3.1
Ex4.1 MIA Page 203
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Starter Questions
S3 Credit
10cm
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Q
6cm
10cm
P
R
7cm
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Angles Triangles
Learning Intention
Success Criteria

1. To use tan of the angle to solve problems.

1. Write down tan ratio.
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2. Use tan of an angle to solve problems.
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Using Tan to calculate angles
S3 Credit
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22
Example
Calculate the tan xo ratio
S3 Credit
P
SOH CAH TOA
Opp
Hyp
Opp
18m
Tan x
Adj
x
Q
18
R
12m
Tan x
Adj
12
1.5
Tan x
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Calculate the size of angle xo
How do we find x?
Tan ?¹is written above
Followed by
2nd
To get this press
Tan
24
2nd
Press
Enter

1.5
Tan ?¹1.5
x
56.3 (1 d.p.)
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Process
1. Identify Hyp, Opp and Adj
2. Write down ratio Tan xo Opp
Adj
3. Calculate xo
2nd
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Now try Exercise 4.2 MIA Page 205
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Starter Questions
S3 Credit
xo
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Angles Triangles
Learning Intention
Success Criteria

1. To use tan of the angle to solve REAL LIFE
   problems.

1. Write down tan ratio.
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2. Use tan of an angle to solve REAL LIFE
problems.
29
Use the tan ratio to find the height h of the
tree to 2 decimal places.
SOH CAH TOA
30
SOH CAH TOA
Example 2
Q1. An aeroplane is preparing to land at Glasgow
Airport. It is over Lennoxtown at present which
is 15km from the airport. The angle of descent
is 6o. What is the height of the plane ?
Aeroplane
c
6o
a 15
Airport
Lennoxtown
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Now try Exercise 5.1 MIA Page 207
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Starter Questions
S3 Credit
xo
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33
Angles Triangles
Learning Intention
Success Criteria

1. To use tan of the angle to find adjacent length.

1. Write down tan ratio.
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2. Use tan of an angle to solve find adjacent
length.
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Use the tan ratio to calculate how far the ladder
is away from the building.
SOH CAH TOA
d m
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Example 2
Q1. An aeroplane is preparing to land at Glasgow
Airport. It is over Lennoxtown at present. It is
at a height of 1.58 km above the ground. It s
angle of descent is 6o. How far is it from the
airport to Lennoxtown?
SOH CAH TOA
Aeroplane
a 1.58 km
6o
Airport
Lennoxtown
36
Now try Exercise 5.2 MIA Page 210
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Starter Questions
S3 Credit
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38
Angles Triangles
Learning Intention
Success Criteria

1. To show how to find an angle using tan ratio.

1. Write down tan ratio.
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2. Use tan ratio to find an angle.
39
Use the tan ratio to calculate the angle that the
support wire makes with the ground.
SOH CAH TOA
4 m
40
Use the tan ratio to find the angle of take-off.
SOH CAH TOA
500 m
41
Now try Exercise 6.1 MIA Page 211
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Starter Questions
S3 Credit
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43
Angles Triangles
Learning Intention
Success Criteria

1. Definite the sine ratio and show how to find an
   angle using this ratio.

1. Write down sine ratio.
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2. Use sine ratio to find an angle.
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The Sine Ratio
S3 Credit
Opp
Sin x
Hyp
Opposite
hypotenuse
x
45
Example
Find the height h
S3 Credit
Hyp
11cm
h
Opp
Opp
Sin x
34
Hyp
h
Sin 34
SOH CAH TOA
11
h
11 x Sin 34
h
11 x Sin 34
6.2cm (1 d.p.)
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Using Sin to calculate angles
S3 Credit
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47
Example
Find the xo
S3 Credit
Hyp
9m
6m
Opp
x
Opp
Sin x
Hyp
SOH CAH TOA
6
Sin x
9
0.667 (3 d.p.)
Sin x
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How do we find x?
Sin ?¹is written above
Sin
Followed by
2nd
To get this press
49
Press
2nd
Enter
0.667

x
Sin ?¹0.667
41.8 (1 d.p.)
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Now try Exercise 7.1 MIA Page 212
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Starter Questions
S3 Credit
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52
Angles Triangles
Learning Intention
Success Criteria

• To show how to use the sine ratio to solve
• REAL-LIFE problems.

1. Write down sine ratio.
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• Use sine ratio to solve
• REAL-LIFE problems.

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The support rope is 11.7m long. The angle between
the rope and ground is 70o. Use the sine ratio to
calculate the height of the flag pole.
SOH CAH TOA
11.7m
54
Use the sine ratio to find the angle of the ramp.
SOH CAH TOA
20 m
55
Now try Exercise 7.2 MIA Page 214
56
Starter Questions
S3 Credit
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57
Angles Triangles
Learning Intention
Success Criteria

1. To show how to calculate the hypotenuse using the
   sine ratio.

1. Write down sine ratio.
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2. Use sine ratio to find the hypotenuse.
58
Example
S3 Credit
SOH CAH TOA
A road AB is right angled at B. The road BC is 5
km. Calculate the length of the new road AC.
Opp
Sin x
Hyp
A
B
5
72
Sin 72
r
5km
r
r
C
r
5.3 km
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Now try Exercise 8.1 MIA Page 215
60
Starter Questions
S3 Credit
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61
Angles Triangles
Learning Intention
Success Criteria

1. Definite the cosine ratio and show how to find an
   length or angle using this ratio.

1. Write down cosine ratio.
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2. Use cosine ratio to find a length or angle.
62
The Cosine Ratio
S3 Credit
Adj
Cos x
Hyp
hypotenuse
x
Adjacent
63
Example
Find the adjacent length b
S3 Credit
b
Adj
40
Adj
Cos x
Opp
Hyp
Hyp
35mm
b
Cos 40
SOH CAH TOA
35
b
35 x Cos 40
b
35 x Cos 40
26.8mm (1 d.p.)
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Using Cos to calculate angles
S3 Credit
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65
Example
Find the angle xo
S3 Credit
Adj
34cm
x
Adj
Cos x
Opp
Hyp
Hyp
45cm
34
Cos x
45
SOH CAH TOA
0.756 (3 d.p.)
Cos x
x
Cos ?¹0.756
41
66
Now try Exercise 9.1 MIA Page 216
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Starter Questions
S3 Credit
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xo
10
6
8
68
The Three Ratios
S3 Credit
adjacent
opposite
Tangent
Cosine
Sine
hypotenuse
adjacent
Sine
adjacent
Cosine
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Cosine
Tangent
hypotenuse
opposite
opposite
Sine
Sine
hypotenuse
69
S3 Credit
CAH
TOA
SOH
70
S3 Credit
Process
1. Write down
SOH CAH TOA
2.
Identify what you want to find
3.
what you know
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Past Paper Type Questions
S3 Credit
SOH CAH TOA
72
Past Paper Type Questions
S3 Credit
SOH CAH TOA
(4 marks)
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Past Paper Type Questions
S3 Credit
SOH CAH TOA
74
Past Paper Type Questions
S3 Credit
SOH CAH TOA
4 marks
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Past Paper Type Questions
S3 Credit
SOH CAH TOA
76
Past Paper Type Questions
S3 Credit
SOH CAH TOA
(4marks)
77
Past Paper Type Questions
S3 Credit
SOH CAH TOA
78
Past Paper Type Questions
S3 Credit
SOH CAH TOA
(4marks)
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Now try Exercise 10.1 10.2 MIA Page 218