Transcript Introduction to Trigonometry

Trigonometry
www.mathsrevision.com
Let’s Investigate
The Tangent Ratio
The Tangent Angle
The Sine Ratio
The Sine Angle
The Cosine Ratio
The Cosine Angle
Mixed Problems
Extension
www.mathsrevision.com
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
www.mathsrevision.com
Trigonometry
Let’s Investigate!
www.mathsrevision.com
Trigonometry means “triangle” and
“measurement”.
We will be using right-angled triangles.
Opposite
www.mathsrevision.com
Trigonometry
x°
Adjacent
Mathemagic!
Opposite
www.mathsrevision.com
Trigonometry
30°
Adjacent
Opposite
= 0.6
Adjacent
Try another!
Opposite
www.mathsrevision.com
Trigonometry
45°
Adjacent
Opposite
= 1
Adjacent
www.mathsrevision.com
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
www.mathsrevision.com
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!
www.mathsrevision.com
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!
www.mathsrevision.com
Trigonometry
What’s the point of all this???
Don’t worry, you’re about to find out!
www.mathsrevision.com
Trigonometry
How high is the tower?
Opp
60°
12 m
Opposite
www.mathsrevision.com
Trigonometry
Copy this!
60°
12 m
Adjacent
www.mathsrevision.com
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!
www.mathsrevision.com
Trigonometry
?
20.8m
So the tower’s
20.8 m high!
Don’t worry, you’ll
be trying plenty of
examples!!
www.mathsrevision.com
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.
www.mathsrevision.com
3cm
Opp
Tan x° =
Adj
Opposite
www.mathsrevision.com
Trigonometry
x°
Adjacent
Example
www.mathsrevision.com
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.)
www.mathsrevision.com
Trigonometry
Now try
Exercise 1.
(HSDU Support Materials)
Starter Questions
www.mathsrevision.com
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.
www.mathsrevision.com
Using Tan to calculate angles
www.mathsrevision.com
Example
www.mathsrevision.com
Trigonometry
Op
p
18m
x°
12m
SOH CAH TOA
Opp
Tan x° =
Adj
Tan x° =
18
12
Tan x° = 1.5
?
Trigonometry
www.mathsrevision.com
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
www.mathsrevision.com
Tan x° = 1.5
Press
2nd
Enter 1.5
Tan ⁻¹
Tan
=
x = Tan ⁻¹1.5 = 56.3° (1 d.p.)
www.mathsrevision.com
Trigonometry
Now try
Exercise 2.
(HSDU Support Materials)
Starter Questions
www.mathsrevision.com
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, ...., ...., ....
www.mathsrevision.com
Trigonometry
Sin x° =
Opposite
www.mathsrevision.com
The Sine Ratio
x°
Opp
Hyp
Example
www.mathsrevision.com
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.)
www.mathsrevision.com
Trigonometry
Now try
Exercise 3.
(HSDU Support Materials)
www.mathsrevision.com
Starter Questions
1. 320  8
2. Calculate 20% of 360
3. Calculate 72 - 58
4. Calculate the value of the
missing angle.
www.mathsrevision.com
57o
Using Sin to calculate angles
www.mathsrevision.com
Example
Trigonometry
www.mathsrevision.com
6m
Op
p
9m
SOH CAH TOA
x°
Opp
Sin x° =
Hyp
6
Sin x° =
9
Sin x° = 0.667 (3 d.p.)
?
Trigonometry
www.mathsrevision.com
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
www.mathsrevision.com
Sin x° = 0.667 (3 d.p.)
Press
2nd
Enter 0.667
Sin ⁻¹
Sin
=
x = Sin ⁻¹0.667 = 41.8° (1 d.p.)
www.mathsrevision.com
Trigonometry
Now try
Exercise 4.
(HSDU Support Materials)
www.mathsrevision.com
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
www.mathsrevision.com
The Cosine Ratio
www.mathsrevision.com
Trigonometry
Cos x° =
x°
Adjacent
Adj
Hyp
Example
Trigonometry
Adj
Cos x° =
Hyp
40°
Op
p
www.mathsrevision.com
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.)
www.mathsrevision.com
Trigonometry
Now try
Exercise 5.
(HSDU Support Materials)
www.mathsrevision.com
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.
www.mathsrevision.com
Using Cos to calculate angles
www.mathsrevision.com
Example
Trigonometry
Adj
Cos x° =
Hyp
Cos x° =
Op
p
www.mathsrevision.com
SOH CAH TOA
34
45
34cm
x°
45cm
Cos x° = 0.756 (3 d.p.)
x = Cos ⁻¹0.756 =40.9° (1 d.p.)
www.mathsrevision.com
Trigonometry
Now try
Exercise 6.
(HSDU Support Materials)
www.mathsrevision.com
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 180 o and 360o
www.mathsrevision.com
The Three Ratios
Sine
www.mathsrevision.com
Cosine
Tangent
Sine
Sine
Tangent
Cosine
Cosine
Sine
www.mathsrevision.com
Trigonometry
www.mathsrevision.com
The Three Ratios
Sin x° =
Opp
Hyp
Cos x° =
Adj
Hyp
Tan x° =
Opp
Adj
www.mathsrevision.com
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°
www.mathsrevision.com
Sin 36°
Sin 30°
Tan 27°
Sin 60°
Tan 40°
Cos 12°
Cos 79°
Sin 35°
www.mathsrevision.com
Example 1
Trigonometry
www.mathsrevision.com
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
www.mathsrevision.com
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
www.mathsrevision.com
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.)
www.mathsrevision.com
Trigonometry
Now try
Exercise 7.
(HSDU Support Materials)
www.mathsrevision.com
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 0 o and 90o
www.mathsrevision.com
www.mathsrevision.com
Extension
www.mathsrevision.com
Example 1
www.mathsrevision.com
Trigonometry
23cm
Op
p
b
SOH CAH TOA
30°
Opp
Sin x° =
Hyp
23
Sin 30° =
b
?
www.mathsrevision.com
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
www.mathsrevision.com
SOH CAH TOA
p=
10.9m (1 d.p.)
Example 3
www.mathsrevision.com
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.)