Transcript 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°
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