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Presents
Let’s Investigate
Extension
The Tangent ratio
The Sine ratio
The Cosine ratio
The three ratios
Let’s Investigate!
Trigonometry means “triangle” and “measurement”.
AdjacentO
pp
osit
e
xx°°
hypotenuse
We will be using right-angled triangles.
30°
Adjacent
Op
posit
e
hypotenuse
OppositeAdjacent
= 0.6
Mathemagic!
45°
Adjacent
Op
posit
e
hypotenuse
OppositeAdjacent
= 1
Try another!
For an angle of 30°, OppositeAdjacent
= 0.6
We write tan 30° = 0.6
Opposite
Adjacent is called the tangent of an angle.
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!
The ancient Greeks discovered this and repeated this for all possible angles.
Now-a-days we can use calculators instead of tables to find the Tan of an angle.
TanOn your calculator press
Notice that your calculator is incredibly accurate!!
Followed by 30, and press
=
Accurate to 9 decimal places!
What’s the point of all this???
Don’t worry, you’re about to find out!
12 m
How high is the tower?
h
60°
60°
12 mAdjacent
Op
posit
e
hypotenuseh
Copy this!
Tan x° =
Opp
Adj
Tan 60° =
h12
= h
12 x Tan 60°h =
12 x Tan 60°= 20.8m (1 d.p.)
Change side, change sign!
Copy this!
So the tower’s 20.8 m high!
Don’t worry, you’ll be trying plenty of examples!!
20.8m
?
The Tangent Ratio
xx°°
Tan x° =
Op
posit
e
Opp
Adjacent
Adj
Example
6565°°
Tan x° =
Opp
Opp
AdjHyp cc
8m8mTan 65° =
c8
= c
8 x Tan 65°
c =
8 x Tan 65° = 17.2m (1 d.p.)
Adj
Change side, change sign!
Now tryExercise 1.
(HSDU Support Materials)
Using Tan to calculate angles
Example
xx°°Tan x° =
Opp
Opp
Adj
Hyp SOH CAH TOA
12m12m
Tan x° = 1812
= 1.5 Tan x°
Adj
18m18m
?
= 1.5Tan x°How do we find x°?
We need to use Tan ⁻¹on the calculator.
2nd
Tan ⁻¹is written above Tan
Tan ⁻¹
To get this press
TanFollowed by
x =
Tan ⁻¹1.5 = 56.3° (1 d.p.)
= 1.5Tan x°
2nd Tan
Tan ⁻¹
Press
Enter =1.5
Now tryExercise 2.
(HSDU Support Materials)
The Sine Ratio
xx°°
Sin x° =
Op
posit
e
OppHyp
hypotenuse
Example
3434°°
Sin x° =
Opp
Opp
Hyp
Hyphh
11c11cmm
Sin 34° =
h11
= h
11 x Sin 34°
h =
11 x Sin 34°
= 6.2cm (1 d.p.)
Change side, change sign!
Now tryExercise 3.
(HSDU Support Materials)
Using Sin to calculate angles
Example
xx°°
Sin x° =
Opp
Opp
Hyp
Hyp
SOH CAH TOA6m6m 9m9m
Sin x° =
69
= 0.667 (3 d.p.)
Sin x°
?
=0.667 (3 d.p.)Sin x°
How do we find x°?
We need to use Sin ⁻¹on the calculator.
2nd
Sin ⁻¹is written above Sin
Sin ⁻¹
To get this press
SinFollowed by
x =
Sin ⁻¹0.667 = 41.8° (1 d.p.)
= 0.667 (3 d.p.)
Sin x°
2nd Sin
Sin ⁻¹
Press
Enter =0.667
Now tryExercise 4.
(HSDU Support Materials)
The Cosine Ratio
Cos x° =
Adjacent
Adj
xx°°
Hyp
hypotenuse
Example4040°°
Cos x° =
Opp
Adj
Hyp
Hyp
bb
35m35mmm
Cos 40° =
b35
= b
35 x Cos 40°
b =
35 x Cos 40°
= 26.8mm (1 d.p.)
Adj
Change side, change sign!
Now tryExercise 5.
(HSDU Support Materials)
Using Cos to calculate angles
Examplexx°°
Cos x° =
Opp
Adj
Hyp
Hyp SOH CAH TOA45cm45cm
Cos x° = 3445
= 0.756 (3 d.p.)Cos x°
x =
Cos ⁻¹0.756 =40.9° (1 d.p.)
Adj34cm34cm
Now tryExercise 6.
(HSDU Support Materials)
The Three Ratios
Cosine
Sine
Tangent
Sine
Sine
Tangent
Cosine
Cosine
Sine
The Ratios
Sin x° =Opp
HypCos x° =
Adj
HypTan x° =
Opp
Adj
The Ratios
Sin x° =Opp
HypCos x° =
Adj
HypTan x° =
Opp
Adj
CAH TOASOH
AC H
OT A
OS H
Copy this!
Mixed Examples
Cos 12°
Sin 60°
Tan 27°
Sin 30°
Sin 35°
Tan 40°
Cos 20°
Cos 79°
Sin 36°
Example 1
4040°°
Sin x° =
Opp
Opp
Hyp
Hyp
SOH CAH TOAhh
1515mm
Sin 40° =
h15
= h
15 x Sin 40°
h =
15 x Sin 40°
= 9.6m (1 d.p.)
Change side, change sign!
Example 23535°°
Cos x° =
Opp
Adj
Hyp
Hyp SOH CAH TOA
bb
23cm23cm
Cos 35° =
b23
= b
23 x Cos 35°
b =
23 x Cos 35°= 18.8cm (1 d.p.)
Adj
Change side, change sign!
Example 3
6060°°Tan x° =
Opp
Opp
Adj
Hyp SOH CAH TOAcc
15m15m
Tan 60° =
c15
= c
15 x Tan 60°c =
15 x Tan 60°= 26.0m (1 d.p.)
Adj
Change side, change sign!
Now tryExercise 7.
(HSDU Support Materials)
Extension
Example 1
3030°°
Sin x° =
Opp
Opp
Hyp
Hyp
SOH CAH TOA23cm23cm bb
Sin 30° =
23b
?
Sin 30° =
23b
Change sides, change signs!
Sin 30°23b
=
(This means b = 23 ÷ Sin 30º)
b=
46 cm
Example 25050°°
Cos x° =
Opp
Adj
Hyp
Hyp SOH CAH TOA
7m7m
pp
Cos 50° =
7p
p=
p= 10.9m (1 d.p.)
Adj
Change sides, change signs!
Cos 50°
7
Example 3
5555°°Tan x° =
Opp
Opp
Adj
Hyp SOH CAH TOA9m9m
ddAdj
Tan 55° =
9d
d=
d= 6.3m (1 d.p.)
Change sides, change signs!
Tan 55°
9
© K Hughes 2001
