S3 3 trigonometry USES

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Trigonometry
S3
Credit
The Tangent Ratio
The Tangent using Angle
The Tangent Ratio in Action
The Tangent (The Adjacent side)
The Tangent (Finding Angle)
The Sine of an Angle
The Sine Ration In Action
The Sine ( Finding the Hypotenuse)
The Cosine of an Angle
Mixed Problems

Starter Questions
1. An F1 car can complete a lap in 2 mins.
A lap is 5 miles in length.
Show that the average speed is 150mph
2. The resistance (R) in copper wire is directly
proportion to its length (L) and inversely to
the square of its radius (r).
Write down an formula connecting R, L and r.
S3
Credit

Angles & Triangles
7-Sep-14 Created by Mr. Lafferty Maths Dept.
Learning Intention Success Criteria
2. Work out Tan Ratio.
1. To identify the
hypotenuse, opposite and
adjacent sides in a right
angled triangle.
1. Understand the terms
hypotenuse, opposite and
adjacent in right angled
triangle.

Trigonometry
Let’s Investigate!
S3
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Trigonometry
S3
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Trigonometry means “triangle” and
“measurement”.
We will be using right-angled triangles.
Adjacent
Opposite
x°

Mathemagic!
Trigonometry
S3
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30°
Adjacent
Opposite
Opposite
Adjacent
= 0.6

Try another!
Trigonometry
S3
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45°
Adjacent
Opposite
Opposite
Adjacent
= 1

Trigonometry
S3
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For an angle of 30°,
Opposite
Adjacent
= 0.6
Opposite
Adjacent
is called the tangent of an angle.
We write tan 30° = 0.6

Trigonometry
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.
S3
Credit

Trigonometry
S3
Credit
Now-a-days we can use
calculators instead of tables
to find the Tan of an angle.
On your calculator press Tan
Followed by 30, and press =
Notice that your calculator is
incredibly accurate!!
Accurate to 9 decimal places!

Trigonometry
S3
Credit
What’s the point of all this???
Don’t worry, you’re about to find out!

Trigonometry
S3
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How high is the tower?
12 m
Opp
60°

Trigonometry
S3
Credit
60°
12 m
Adjacent
Opposite
Copy this!

Trigonometry
S3
Credit
Tan x° =
Opp
Adj
Tan 60° =
Opp
12
12 x Tan 60° = Opp
Opp =12 x Tan 60° = 20.8m (1 d.p.)
Copy this!

Trigonometry
S3
Credit
So the tower’s
20.8 m high!
Don’t worry, you’ll
be trying plenty of
examples!!
20.8m

Trigonometry
Adj
Tan x° =
x°
Opposite
Opp
Adjacent
S3
Credit

Example
Trigonometry
65°
Opp
Tan x° =
Opp
Adj
h
8m Tan 65° =
h
8
8 x Tan 65° = h
h = 8 x Tan 65° = 17.2m (1 d.p.)
S3
Credit
Find the
height h
SOH CAH TOA

Trigonometry
S3
Credit
Class Group
Identifying the
Tan Ratio Ex 3.1 & Ex4.1
MIA Page 203

Starter Questions
1. Find the area and perimeter
2. Is the triangle right angled at P.
Explain your answer.
3. Factorise x  x
of the circle.
2 2 3 2
10cm
P
Q
R
6cm
7cm
10cm
S3
Credit

Angles & Triangles
2. Use tan of an angle to solve
problems.
1. To use tan of the angle to
solve problems.
1. Write down tan ratio.

Using Tan to calculate angles
S3
Credit

Trigonometry
SOH CAH TOA
18
12
Example
x°
Tan x° =
Opp
Opp
Adj
P
12m Tan x° =
Tan x° = 1.5
18m
S3
Credit
Calculate the
tan xo ratio
Q
R

Calculate the
Trigonometry
S3
Credit
Tan x° = 1.5
How do we find x°?
size of
angle xo
We need to use Tan ⁻¹on the
calculator.
Tan ⁻¹is written above Tan
2nd
Tan ⁻¹
To get this press Followed by Tan

Trigonometry
S3
Credit
Tan x° = 1.5
Tan ⁻¹
2nd Tan
Press
Enter 1.5 =
x = Tan ⁻¹1.5 = 56.3° (1 d.p.)

Trigonometry
S3
Credit
Process
1. Identify Hyp, Opp and Adj
2. Write down ratio Tan xo = Opp
Adj
Tan ⁻¹
3. Calculate xo 2nd Tan

Trigonometry
S3
Credit
Now try
Exercise 4.2
MIA Page 205

Starter Questions
 
1. True or false 30 5 + 6 4 = 48
2. Write in scientific notation 0.0456
3. Identify the sides of the triangle.
4. The subway train takes 6 mins to travel
between 2 stations 3 miles apar
t.
Show that it's average speed is 30mph.
S3
Credit
xo

Angles & Triangles
REAL LIFE problems.
solve REAL LIFE problems.

Trigonometry
S3
Credit
Use the tan ratio to find the height h of the tree
to 2 decimal places.
SOH CAH TOA
rod
47o
7-Sep-14 Compiled by Mr. Lafferty Maths Dept.
8m
o opp h
tan 47 = =
adj 8
o h
tan 47 =
8
o h = 8× tan 47
h = 8.58m

Trigonometry
S3
Credit
Q1. An aeroplane is preparing to land at Glasgow Airport.
It is over Lennoxtown at present which is 15km from
the airport. The angle of descent is 6o.
6o
7-Sep-14
What is the height of the plane ?
Compiled by Mr. Lafferty Maths
Dept.
Aeroplane
a = 15
c
Airport Lennoxtown
Example 2
o h
tan 6 =
15
o h =15× tan 6
h =1.58km
SOH CAH TOA

Trigonometry
S3
Credit
Now try
Exercise 5.1
MIA Page 207

Starter Questions

1. Explain why 6 + 9 3 = 9 and not 5
2. Write in scientific notation 32.56
3. Identify the sides of the triangle.
4. The train takes 10 mins to travel
between 2 stations 6miles apart.
Find the average speed of the train.
S3
Credit
xo

Angles & Triangles
find adjacent length.
find adjacent length.

Trigonometry
S3
Credit
Use the tan ratio to calculate how far the ladder
is away from the building.
SOH CAH TOA
45o
12m
ladder
o opp 12
tan 45 = =
adj d
o
12
d =
tan 45
d =12m
d m

Trigonometry
S3
Credit
Q1. An aeroplane is preparing to land at Glasgow Airport. It is over
Lennoxtown at present. It is at a height of 1.58 km above the ground. It
‘s angle of descent is 6o.
How far is it from the airport to Lennoxtown?
6o
Aeroplane
a = 1.58 km
Airport Lennoxtown
Example 2
o 1.58
tan 6 =
d
o
1.58
d =
tan 6
d =15 km
SOH CAH TOA

Trigonometry
S3
Credit
Now try
Exercise 5.2
MIA Page 210

Starter Questions
3 3 5
1. +

4 5 7
2. Explain why y =6 when a = (-1) b = 2
y = (a -b)(b- a)(2a -b)
p
Q3. Given T =k .
v
Find k when T =6 P = 18 and v = 9.
S3
Credit

Angles & Triangles
2. Use tan ratio to find an angle.
1. To show how to find an
angle using tan ratio.

Trigonometry
S3
Credit
Use the tan ratio to calculate the angle that the
support wire makes with the ground.
SOH CAH TOA
xo
11m
o opp 11
tan x = =
adj 4
11
4
 
 
 
o -1 x = tan
o o x = 70
4 m

Trigonometry
S3
Credit
Use the tan ratio to find the angle of take-off.
SOH CAH TOA
xo 88m
o opp 88
tan x = =
adj 500
o tan x = 0.176
o -1 o x = tan (0.176) = 10
500 m

Trigonometry
S3
Credit
Now try
Exercise 6.1
MIA Page 211

Starter Questions
1. A train takes 12 minutes to travel
between 2 stations.
Show that the average speed is 60km/hr
if the stations are 6miles apart.
2. Calculate A when w= (-3) y = 4
A = (w - y) + (y - w)
S3
Credit

Angles & Triangles
2. Use sine ratio to find an
angle.
1. Definite the sine ratio and
show how to find an angle
using this ratio.
1. Write down sine ratio.

Trigonometry
The Sine Ratio
Sin x° =
x°
Opposite
Opp
Hyp
S3
Credit

Example
Trigonometry
h 11cm
Opp Opp
Hyp
Sin x° = 34°
Sin 34° =
h
11
11 x Sin 34° = h
h = 11 x Sin 34° = 6.2cm (1 d.p.)
S3
Credit
Find the
height h
SOH CAH TOA

Using Sin to calculate angles
S3
Credit

Example
Trigonometry
x°
6m 9m
Sin x° =
Opp
Opp
Hyp
Sin x° =
6
9
Sin x° = 0.667 (3 d.p.)
S3
Credit
Find the xo
SOH CAH TOA

Trigonometry
S3
Credit
Sin x° =0.667 (3 d.p.)
How do we find x°?
We need to use Sin ⁻¹on the
calculator.
Sin ⁻¹is written above Sin
2nd
Sin ⁻¹
To get this press Followed by Sin

Trigonometry
S3
Credit
Sin x° = 0.667 (3 d.p.)
Sin ⁻¹
2nd Sin
Press
Enter 0.667 =
x = Sin ⁻¹0.667 = 41.8° (1 d.p.)

Trigonometry
S3
Credit
Now try
Exercise 7.1
MIA Page 212

Starter Questions
1 2 3 1. Explain why we can simply pick out Q
the 5 figure summary for the data
19, 15, 11, 22, 9, 12, 11
2. Show that the original price of a car is £9000
If it cos
, Q and Q
then find
ts £8100 after a discount of 10%
3. A lorry is travelling at 40mph.
It has travelled 60 miles.
How long has it taken to travel 60 miles.
S3
Credit

Angles & Triangles
2. Use sine ratio to solve
REAL-LIFE problems.
1. To show how to use the
sine ratio to solve
REAL-LIFE problems.

Trigonometry
S3
Credit
The support rope is 11.7m long. The angle between
the rope and ground is 70o. Use the sine ratio to
calculate the height of the flag pole.
SOH CAH TOA
70o
h
o opp h
sin 70 = =
hyp 11.7
h  o =11.7 sin70
h =11 m
11.7m

Trigonometry
S3
Credit
Use the sine ratio to find the angle of the ramp.
SOH CAH TOA
20 m
xo 10m
o opp 10
sin x = =
hyp 20
o 10
sin x =
20
o -1 o 10
x = sin = 30
 
 
 20


Trigonometry
S3
Credit
Now try
Exercise 7.2
MIA Page 214

Starter Questions
1. Fill in the ? marks.
x  x x x
2. Calculate A when w= (-10)
2 2 9 7 = (2 ?)( ?)
2
A =
w
4
S3
Credit

Angles & Triangles
2. Use sine ratio to find the
hypotenuse.
1. To show how to calculate
the hypotenuse using the
sine ratio.

Example
Trigonometry
SOH CAH TOA
72°
Sin x° =
Opp
Hyp
Sin 72° =
5
r
5 r
r =
r = 5.3 km
5km
S3
Credit
B A
C
sin72o
A road AB is right angled at B.
The road BC is 5 km.
Calculate the length of
the new road AC.

Trigonometry
S3
Credit
Now try
Exercise 8.1
MIA Page 215

Starter Questions
1 3 1. Explain why we can simply pick out Q
then find the 5 figure summary for the data
9, 5, 11, 2, 9, 2
2. Find the original price of a football
If it costs £20 after a discoun
and Q
t of 80%
3. A lorry is travelling at 50mph.
It has travelled 75 miles.
Show that the time taken is 1hr 30 mins.
S3
Credit

Angles & Triangles
2. Use cosine ratio to find a
length or angle.
1. Definite the cosine ratio
and show how to find an
length or angle using this
ratio.
1. Write down cosine ratio.

The Cosine Ratio
Trigonometry
Cos x° =
Adjacent
Adj
x°
Hyp
S3
Credit

Trigonometry
SOH CAH TOA
Example
40°
Cos x° =
Opp
Adj
Hyp
b
35mm
Cos 40° =
b
35
35 x Cos 40° = b
b = 35 x Cos 40°= 26.8mm (1 d.p.)
S3
Credit
Find the
adjacent
length b

Using Cos to calculate angles
S3
Credit

Trigonometry
34cm
SOH CAH TOA
Example
x°
Cos x° =
Opp
Adj
Hyp
45cm
Cos x° =
34
45
Cos x° = 0.756 (3 d.p.)
x = Cos ⁻¹0.756 =41°
S3
Credit
Find the
angle xo

Trigonometry
S3
Credit
Now try
Exercise 9.1
MIA Page 216

Starter Questions
o
1. Calculate 104 x 100
putting your answer in standard form.
2. Is this triangle right angled ?
If yes, find the size of angle x .
If no find the area of the triangle.
S3
Credit
xo
6
8
10

The Three Ratios
Cosine
adjacent
hypotenuse
Cosine
Tangent
Sine
Sine
Sine
Tangent
Cosine
Sine
S3
Credit
opposite
opposite opposite
adjacent
adjacent
hypotenuse
hypotenuse

Trigonometry
Sin x° =
Opp
Hyp
Cos x° =
Adj
Hyp
Tan x° =
Opp
Adj
SOHCAH TOA
S3
Credit

Trigonometry
Process
SOH CAH TOA
Copy this!
S3
Credit
1. Write down
Identify what you want to find
2.
3. what you know

Trigonometry
Past Paper Type Questions
S3
Credit
SOH CAH TOA

Trigonometry
S3
Credit
(4 marks)
SOH CAH TOA

Trigonometry
S3
Credit
SOH CAH TOA
4 marks

Trigonometry
S3
Credit
(4marks)
SOH CAH TOA

Trigonometry
S3
Credit
SOH CAH TOA
(4marks)

Trigonometry

Trigonometry
S3
Credit
Now try
Exercise 10.1 & 10.2
MIA Page 218

S3 3 trigonometry USES

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