Transcript Introduction to Trigonometry

Trigonometry
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Let’s Investigate
The Tangent Ratio
The Tangent Angle
The Sine Ratio
The Sine Angle
The Cosine Ratio
The Cosine Angle
Mixed Problems
Extension
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Starter Questions
1. Find the missing value
3 ?
=
4 20
2. Calculate 20% of 6000
3. What is the next three numbers
in the sequence
9, 15, 21, 27, ...., ...., ....
4. Round 72 to the nearest 10
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Trigonometry
Let’s Investigate!
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Trigonometry means “triangle” and
“measurement”.
We will be using right-angled triangles.
Opposite
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Trigonometry
x°
Adjacent
Mathemagic!
Opposite
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Trigonometry
30°
Adjacent
Opposite
= 0.6
Adjacent
Try another!
Opposite
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Trigonometry
45°
Adjacent
Opposite
= 1
Adjacent
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Trigonometry
For an angle of 30°,
Opposite
= 0.6
Adjacent
Opposite
is called the tangent of an angle.
Adjacent
We write tan 30° = 0.6
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Trigonometry
The ancient Greeks
discovered this and
repeated this for
possible angles.
Tan 25°
0.466
Tan 26°
0.488
Tan 27°
0.510
Tan 28°
0.532
Tan 30° =0.554
0.577
Tan 29°
Tan 30°
0.577
Tan 31°
0.601
Tan 32°
0.625
Tan 33°
0.649
Tan 34°
0.675
Accurate to
3 decimal places!
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Trigonometry
Now-a-days we can use
calculators instead of tables
to find the Tan of an angle.
On your calculator press
Followed by 30, and press
Tan
=
Notice that your calculator is
incredibly accurate!!
Accurate to 9 decimal places!
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Trigonometry
What’s the point of all this???
Don’t worry, you’re about to find out!
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Trigonometry
How high is the tower?
Opp
60°
12 m
Opposite
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Trigonometry
Copy this!
60°
12 m
Adjacent
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Trigonometry
Opp
Tan x° =
Adj
Change side,
change sign!
Opp
Tan 60° =
12
12 x Tan 60° = Opp
Opp =12 x Tan 60° = 20.8m (1 d.p.)
Copy this!
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Trigonometry
?
20.8m
So the tower’s
20.8 m high!
Don’t worry, you’ll
be trying plenty of
examples!!
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Starter Questions
1. Find the perimeter of the shape
2. Calculate 30% of 900
3. Find the area of the rectangle
6cm in length by 4 cm wide.
4. Name the shape.
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3cm
Opp
Tan x° =
Adj
Opposite
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Trigonometry
x°
Adjacent
Example
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Trigonometry
Op
c p
Opp
Tan x° =
Adj
65°
8m
Tan 65° =
c
8
Change side,
change sign!
8 x Tan 65° = c
c = 8 x Tan 65° = 17.2m (1 d.p.)
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Trigonometry
Now try
Exercise 1.
(HSDU Support Materials)
Starter Questions
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1. Name the part of the circle.
2. Calculate 60% of 300
3. If I am facing North and turn 90o
clockwise, which direction am I facing
4. How many lines of symmetry
has the shape.
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Using Tan to calculate angles
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Example
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Trigonometry
Op
p
18m
x°
12m
SOH CAH TOA
Opp
Tan x° =
Adj
Tan x° =
18
12
Tan x° = 1.5
?
Trigonometry
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Tan x° = 1.5
How do we find x°?
We need to use Tan ⁻¹on the
calculator.
Tan ⁻¹is written above
To get this press
2nd
Tan ⁻¹
Tan
Followed by
Tan
Trigonometry
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Tan x° = 1.5
Press
2nd
Enter 1.5
Tan ⁻¹
Tan
=
x = Tan ⁻¹1.5 = 56.3° (1 d.p.)
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Trigonometry
Now try
Exercise 2.
(HSDU Support Materials)
Starter Questions
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1.
13.9 x 7
2. Calculate 23.34 x 10
3
3.
of 80
4
4. Find the missing number
1, 1, 2, 3, 5, 8, ...., ...., ....
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Trigonometry
Sin x° =
Opposite
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The Sine Ratio
x°
Opp
Hyp
Example
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Trigonometry
O
Op
p
Opp
Sin x° =
Hyp
Sin 34° =
O
11
11cm
34°
Change side, change sign!
11 x Sin 34° = O
O = 11 x Sin 34° = 6.2cm (1 d.p.)
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Trigonometry
Now try
Exercise 3.
(HSDU Support Materials)
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Starter Questions
1. 320  8
2. Calculate 20% of 360
3. Calculate 72 - 58
4. Calculate the value of the
missing angle.
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57o
Using Sin to calculate angles
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Example
Trigonometry
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6m
Op
p
9m
SOH CAH TOA
x°
Opp
Sin x° =
Hyp
6
Sin x° =
9
Sin x° = 0.667 (3 d.p.)
?
Trigonometry
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Sin x° =0.667
(3 d.p.)
How do we find x°?
We need to use Sin ⁻¹on the
calculator.
Sin ⁻¹is written above
To get this press
2nd
Sin ⁻¹
Sin
Followed by
Sin
Trigonometry
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Sin x° = 0.667 (3 d.p.)
Press
2nd
Enter 0.667
Sin ⁻¹
Sin
=
x = Sin ⁻¹0.667 = 41.8° (1 d.p.)
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Trigonometry
Now try
Exercise 4.
(HSDU Support Materials)
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Starter Questions
1. 2.39 - 1.58 + 3.2
2. Calculate 15% of 380
3. What is the next three numbers
in the sequence
2, 15, 28, 41, ...., ...., ....
4. Round 3.25 to the nearest 0.1
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The Cosine Ratio
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Trigonometry
Cos x° =
x°
Adjacent
Adj
Hyp
Example
Trigonometry
Adj
Cos x° =
Hyp
40°
Op
p
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b
b
Cos 40° =
35
35mm
Change side, change sign!
35 x Cos 40° = b
b = 35 x Cos 40°= 26.8mm (1 d.p.)
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Trigonometry
Now try
Exercise 5.
(HSDU Support Materials)
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Starter Questions
Q1.
Calculate 75% of £200
Q2.
Round to 1 decimal place 2.354.
Q3.
How many minutes in 3hours
Q4.
The answer to the question is 180. What is the
question.
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Using Cos to calculate angles
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Example
Trigonometry
Adj
Cos x° =
Hyp
Cos x° =
Op
p
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SOH CAH TOA
34
45
34cm
x°
45cm
Cos x° = 0.756 (3 d.p.)
x = Cos ⁻¹0.756 =40.9° (1 d.p.)
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Trigonometry
Now try
Exercise 6.
(HSDU Support Materials)
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Starter Questions
1. Calculate 14 x 100
2. What kind of angle is this
3. 56.98  10
4. Name the angle that is between 180o and 360o
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The Three Ratios
Sine
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Cosine
Tangent
Sine
Sine
Tangent
Cosine
Cosine
Sine
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Trigonometry
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The Three Ratios
Sin x° =
Opp
Hyp
Cos x° =
Adj
Hyp
Tan x° =
Opp
Adj
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Trigonometry
Sin x° =
Opp
Hyp
O
S H
O
S H
Cos x° =
Adj
Hyp
A
C H
A
C H
Copy this!
Tan x° =
Opp
Adj
O
T A
O
T A
Mixed Examples
Cos 20°
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Sin 36°
Sin 30°
Tan 27°
Sin 60°
Tan 40°
Cos 12°
Cos 79°
Sin 35°
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Example 1
Trigonometry
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SOH CAH TOA
Opp
Sin x° =
Hyp
O
Sin 40° =
15
O
Op
p
15m
40°
Change side, change sign!
15 x Sin 40° = O
O=
15 x Sin 40° = 9.6m (1 d.p.)
Example 2
Trigonometry
Adj
Cos x° =
Hyp
b
Cos 35° =
23
b
35°
Op
p
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SOH CAH TOA
23cm
Change side, change sign!
23 x Cos 35° = b
b = 23 x Cos 35° = 18.8cm (1 d.p.)
Example 3
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Trigonometry
Op
c p
60°
15m
SOH CAH TOA
Opp
Tan x° =
Adj
c
Tan 60° =
15
Change side,
change sign!
15 x Tan 60° = c
c = 15 x Tan 60° = 26.0m (1 d.p.)
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Trigonometry
Now try
Exercise 7.
(HSDU Support Materials)
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Level E
Starter Questions
1. Calculate 41.9 x 100
2. What kind of angle is this
3. 1.268  100
4. Name the angle that is between 0o and 90o
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Extension
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Example 1
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Trigonometry
23cm
Op
p
b
SOH CAH TOA
30°
Opp
Sin x° =
Hyp
23
Sin 30° =
b
?
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Trigonometry
23
Sin 30° =
b
Change sides, change signs!
23
b=
Sin 30°
(This means b = 23 ÷ Sin 30º)
b= 46 cm
Example 2
Trigonometry
7m
50°
Adj
Cos x° =
Hyp
p
7
Cos 50° =
Change sides, change signs!
p
7
p=
Cos 50°
Op
p
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SOH CAH TOA
p=
10.9m (1 d.p.)
Example 3
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Trigonometry
Op
9m p
55°
d
SOH CAH TOA
Opp
Tan x° =
Adj
9
Tan 55° =
d
9
d=
Tan 55°
Change sides,
change signs!
d= 6.3m (1 d.p.)