Presents
Let’s Investigate
The Tangent ratio
The Sine ratio
The Cosine ratio
The three ratios
Extension
Let’s Investigate!
Trigonometry means “triangle” and
“measurement”.
We will be using right-angled triangles.
Opposite
x°
Adjacent
Mathemagic!
Opposite
30°
Adjacent
Opposite
= 0.6
Adjacent
Try another!
Opposite
45°
Adjacent
Opposite
= 1
Adjacent
For an angle of 30°,
Opposite
Adjacent
Opposite
= 0.6
Adjacent
is called the tangent of an angle.
We write tan 30° = 0.6
The ancient Greeks discovered this and
repeated this for all possible angles.
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
Tan 30° = 0.577
Accurate to 3 decimal places!
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!
What’s the point of all this???
Don’t worry, you’re about to find out!
How high is the tower?
h
60°
12 m
Copy this!
Opposite
h
60°
12 m
Adjacent
Opp
Tan x° =
Adj
h
Tan 60° =
12
Copy this!
Change side,
change sign!
12 x Tan 60° = h
h = 12 x Tan 60° = 20.8m (1 d.p.)
So the tower’s 20.8 m high!
?
20.8m
Don’t worry, you’ll
be trying plenty of
examples!!
The Tangent Ratio
Opp
Tan x° =
Adj
Opposite
x°
Adjacent
Example
Op
c p
65°
8m
Opp
Tan x° =
Adj
Tan 65° =
c
8
Change side,
change sign!
8 x Tan 65° = c
c = 8 x Tan 65° = 17.2m (1 d.p.)
Now try
Exercise 1.
(HSDU Support Materials)
Using Tan to calculate angles
Example
Op
p
18m
x°
12m
SOH CAH TOA
Opp
Tan x° =
Adj
Tan x° =
18
12
Tan x° = 1.5
?
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
Tan ⁻¹
Tan
2nd
Followed by
Tan
Tan x° = 1.5
Press
2nd
Tan ⁻¹
Tan
Enter 1.5
=
x = Tan ⁻¹1.5 = 56.3° (1 d.p.)
Now try
Exercise 2.
(HSDU Support Materials)
The Sine Ratio
Sin x° =
Opp
Hyp
Opposite
x°
h
Op
p
Example
11cm
34°
Opp
Sin x° =
Hyp
h
Sin 34° =
11
Change side, change sign!
11 x Sin 34° = h
h = 11 x Sin 34° = 6.2cm (1 d.p.)
Now try
Exercise 3.
(HSDU Support Materials)
Using Sin to calculate angles
6m
Op
p
Example
9m
SOH CAH TOA
x°
Opp
Sin x° =
Hyp
6
Sin x° =
9
Sin x° = 0.667 (3 d.p.)
?
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
Sin ⁻¹
Sin
2nd
Followed by
Sin
Sin x° = 0.667 (3 d.p.)
Press
2nd
Sin ⁻¹
Sin
Enter 0.667
=
x = Sin ⁻¹0.667 = 41.8° (1 d.p.)
Now try
Exercise 4.
(HSDU Support Materials)
The Cosine Ratio
Cos x° =
Adj
Hyp
x°
Adjacent
b
40°
Example
Op
35mm
Adj
Cos x° =
Hyp
b
Cos 40° =
35
Change side, change sign!
35 x Cos 40° = b
b = 35 x Cos 40°= 26.8mm (1 d.p.)
Now try
Exercise 5.
(HSDU Support Materials)
Using Cos to calculate angles
34cm
x°
Example
Op
SOH CAH TOA
45cm
Adj
Cos x° =
Hyp
34
Cos x° =
45
Cos x° = 0.756 (3 d.p.)
x = Cos ⁻¹0.756 =40.9° (1 d.p.)
Now try
Exercise 6.
(HSDU Support Materials)
Tangent
Sine
Cosine
The Three Ratios
Sine
Sine
Tangent
Cosine
Cosine
Sine
The Ratios
Sin x° =
Opp
Hyp
Cos x° =
Adj
Hyp
Tan x° =
Opp
Adj
The Ratios
Sin x° =
Opp
Hyp
Cos x° =
Adj
Hyp
Copy this!
Tan x° =
Opp
Adj
O
S H
A
C H
O
T A
SOH
CAH
TOA
Tan 27°
Sin 36°
Cos 20°
Mixed Examples
Sin 30°
Sin 60°
Tan 40°
Cos 12°
Cos 79°
Sin 35°
h
Op
p
Example 1
15m
SOH CAH TOA
40°
Opp
Sin x° =
Hyp
h
Sin 40° =
15
Change side, change sign!
15 x Sin 40° = h
h = 15 x Sin 40° = 9.6m (1 d.p.)
b
35°
Example 2
Op
SOH CAH TOA
23cm
Adj
Cos x° =
Hyp
b
Cos 35° =
23
Change side, change sign!
23 x Cos 35° = b
b = 23 x Cos 35° = 18.8cm (1 d.p.)
Example 3
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.)
Now try
Exercise 7.
(HSDU Support Materials)
Extension
23cm
Op
p
Example 1
b
SOH CAH TOA
30°
Opp
Sin x° =
Hyp
23
Sin 30° =
b
?
23
Sin 30° =
b
Change sides, change signs!
23
b=
Sin 30°
(This means b = 23 ÷ Sin 30º)
b= 46 cm
7m
50°
Example 2
Op
SOH CAH TOA
p
Adj
Cos x° =
Hyp
7
Cos 50° =
Change sides, change signs!
p
7
p=
Cos 50°
p= 10.9m (1 d.p.)
Example 3
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.)
© K Hughes 2001