# Trigonometry in a Right angles Triangle ```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&deg;
Mathemagic!
Opposite
30&deg;
Opposite
= 0.6
Try another!
Opposite
45&deg;
Opposite
= 1
For an angle of 30&deg;,
Opposite
Opposite
= 0.6
is called the tangent of an angle.
We write tan 30&deg; = 0.6
The ancient Greeks discovered this and
repeated this for all possible angles.
Tan 25&deg;
0.466
Tan 26&deg;
0.488
Tan 27&deg;
0.510
Tan 28&deg;
0.532
Tan 29&deg;
0.554
Tan 30&deg;
0.577
Tan 31&deg;
0.601
Tan 32&deg;
0.625
Tan 33&deg;
0.649
Tan 34&deg;
0.675
Tan 30&deg; = 0.577
Accurate to 3 decimal places!
Now-a-days we can use
to find the Tan of an angle.
Followed by 30, and press
Tan
=
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&deg;
12 m
Copy this!
Opposite
h
60&deg;
12 m
Opp
Tan x&deg; =
h
Tan 60&deg; =
12
Copy this!
Change side,
change sign!
12 x Tan 60&deg; = h
h = 12 x Tan 60&deg; = 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&deg; =
Opposite
x&deg;
Example
Op
c p
65&deg;
8m
Opp
Tan x&deg; =
Tan 65&deg; =
c
8
Change side,
change sign!
8 x Tan 65&deg; = c
c = 8 x Tan 65&deg; = 17.2m (1 d.p.)
Now try
Exercise 1.
(HSDU Support Materials)
Using Tan to calculate angles
Example
Op
p
18m
x&deg;
12m
SOH CAH TOA
Opp
Tan x&deg; =
Tan x&deg; =
18
12
Tan x&deg; = 1.5
?
Tan x&deg; = 1.5
How do we find x&deg;?
We need to use Tan ⁻&sup1;on the
calculator.
Tan ⁻&sup1;is written above
To get this press
Tan ⁻&sup1;
Tan
2nd
Followed by
Tan
Tan x&deg; = 1.5
Press
2nd
Tan ⁻&sup1;
Tan
Enter 1.5
=
x = Tan ⁻&sup1;1.5 = 56.3&deg; (1 d.p.)
Now try
Exercise 2.
(HSDU Support Materials)
The Sine Ratio
Sin x&deg; =
Opp
Hyp
Opposite
x&deg;
h
Op
p
Example
11cm
34&deg;
Opp
Sin x&deg; =
Hyp
h
Sin 34&deg; =
11
Change side, change sign!
11 x Sin 34&deg; = h
h = 11 x Sin 34&deg; = 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&deg;
Opp
Sin x&deg; =
Hyp
6
Sin x&deg; =
9
Sin x&deg; = 0.667 (3 d.p.)
?
Sin x&deg; =0.667
(3 d.p.)
How do we find x&deg;?
We need to use Sin ⁻&sup1;on the
calculator.
Sin ⁻&sup1;is written above
To get this press
Sin ⁻&sup1;
Sin
2nd
Followed by
Sin
Sin x&deg; = 0.667 (3 d.p.)
Press
2nd
Sin ⁻&sup1;
Sin
Enter 0.667
=
x = Sin ⁻&sup1;0.667 = 41.8&deg; (1 d.p.)
Now try
Exercise 4.
(HSDU Support Materials)
The Cosine Ratio
Cos x&deg; =
Hyp
x&deg;
b
40&deg;
Example
Op
35mm
Cos x&deg; =
Hyp
b
Cos 40&deg; =
35
Change side, change sign!
35 x Cos 40&deg; = b
b = 35 x Cos 40&deg;= 26.8mm (1 d.p.)
Now try
Exercise 5.
(HSDU Support Materials)
Using Cos to calculate angles
34cm
x&deg;
Example
Op
SOH CAH TOA
45cm
Cos x&deg; =
Hyp
34
Cos x&deg; =
45
Cos x&deg; = 0.756 (3 d.p.)
x = Cos ⁻&sup1;0.756 =40.9&deg; (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&deg; =
Opp
Hyp
Cos x&deg; =
Hyp
Tan x&deg; =
Opp
The Ratios
Sin x&deg; =
Opp
Hyp
Cos x&deg; =
Hyp
Copy this!
Tan x&deg; =
Opp
O
S H
A
C H
O
T A
SOH
CAH
TOA
Tan 27&deg;
Sin 36&deg;
Cos 20&deg;
Mixed Examples
Sin 30&deg;
Sin 60&deg;
Tan 40&deg;
Cos 12&deg;
Cos 79&deg;
Sin 35&deg;
h
Op
p
Example 1
15m
SOH CAH TOA
40&deg;
Opp
Sin x&deg; =
Hyp
h
Sin 40&deg; =
15
Change side, change sign!
15 x Sin 40&deg; = h
h = 15 x Sin 40&deg; = 9.6m (1 d.p.)
b
35&deg;
Example 2
Op
SOH CAH TOA
23cm
Cos x&deg; =
Hyp
b
Cos 35&deg; =
23
Change side, change sign!
23 x Cos 35&deg; = b
b = 23 x Cos 35&deg; = 18.8cm (1 d.p.)
Example 3
Op
c p
60&deg;
15m
SOH CAH TOA
Opp
Tan x&deg; =
c
Tan 60&deg; =
15
Change side,
change sign!
15 x Tan 60&deg; = c
c = 15 x Tan 60&deg; = 26.0m (1 d.p.)
Now try
Exercise 7.
(HSDU Support Materials)
Extension
23cm
Op
p
Example 1
b
SOH CAH TOA
30&deg;
Opp
Sin x&deg; =
Hyp
23
Sin 30&deg; =
b
?
23
Sin 30&deg; =
b
Change sides, change signs!
23
b=
Sin 30&deg;
(This means b = 23 &divide; Sin 30&ordm;)
b= 46 cm
7m
50&deg;
Example 2
Op
SOH CAH TOA
p
Cos x&deg; =
Hyp
7
Cos 50&deg; =
Change sides, change signs!
p
7
p=
Cos 50&deg;
p= 10.9m (1 d.p.)
Example 3
Op
9m p
55&deg;
d
SOH CAH TOA
Opp
Tan x&deg; =