Transcript Introduction to Trigonometry

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Trigonometry
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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
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Starter Questions
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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.
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Angles & Triangles
Learning Intention
1. To identify the
hypotenuse, opposite and
adjacent sides in a right
angled triangle.
Success Criteria
1. Understand the terms
hypotenuse, opposite and
adjacent in right angled
triangle.
2. Work out Tan Ratio.
16-Jul-15
Created by Mr. Lafferty Maths Dept.
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Trigonometry
Let’s Investigate!
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Trigonometry means “triangle” and
“measurement”.
We will be using right-angled triangles.
Opposite
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Trigonometry
x°
Adjacent
Mathemagic!
Trigonometry
Opposite
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30°
Adjacent
Opposite
= 0.6
Adjacent
Try another!
Trigonometry
Opposite
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45°
Adjacent
Opposite
= 1
Adjacent
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Trigonometry
For an angle of 30°,
Opposite
= 0.6
Adjacent
Opposite
is called the tangent of an angle.
Adjacent
We write tan 30° = 0.6
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Trigonometry
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 30° =0.554
0.577
Tan 29°
Tan 30°
0.577
Tan 31°
0.601
Tan 32°
0.625
Tan 33°
0.649
Tan 34°
0.675
Accurate to
3 decimal places!
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Trigonometry
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!
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Trigonometry
What’s the point of all this???
Don’t worry, you’re about to find out!
Trigonometry
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How high is the tower?
Opp
60°
12 m
Trigonometry
Opposite
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Copy this!
60°
12 m
Adjacent
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Trigonometry
Opp
Tan x° =
Adj
Opp
Tan 60° =
12
12 x Tan 60° = Opp
Opp =12 x Tan 60° = 20.8m (1 d.p.)
Copy this!
Trigonometry
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So the tower’s
20.8 m high!
20.8m
Don’t worry, you’ll
be trying plenty of
examples!!
Opp
Tan x° =
Adj
Opposite
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Trigonometry
x°
Adjacent
Example
Trigonometry
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Find the
height h
h
65°
8m
Opp
SOH CAH TOA
Opp
Tan x° =
Adj
Tan 65° =
h
8
8 x Tan 65° = h
h = 8 x Tan 65° = 17.2m (1 d.p.)
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Trigonometry
Class Group
Identifying the
Tan Ratio Ex 3.1 & Ex4.1
MIA Page 203
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Starter Questions
1. Find the area and perimeter
10cm
of the circle.
2. Is the triangle right angled at P.
Q
Explain your answer.
6cm
3. Factorise 2x2  3x  2
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10cm
P
7cm
R
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Angles & Triangles
Learning Intention
1. To use tan of the angle to
solve problems.
Success Criteria
1. Write down tan ratio.
2. Use tan of an angle to solve
problems.
16-Jul-15
Created by Mr. Lafferty Maths Dept.
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Using Tan to calculate angles
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Example
Trigonometry
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Calculate the
tan xo ratio
P
SOH CAH TOA
Opp
18m
R
x°
12m
Q
Opp
Tan x° =
Adj
Tan x° =
18
12
Tan x° = 1.5
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Calculate the
size of
angle xo
Trigonometry
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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
2nd
Tan ⁻¹
Tan
Followed by
Tan
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Trigonometry
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Tan x° = 1.5
Press
2nd
Enter 1.5
Tan ⁻¹
Tan
=
x = Tan ⁻¹1.5 = 56.3° (1 d.p.)
Trigonometry
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Process
1.
Identify Hyp, Opp and Adj
2. Write down ratio Tan xo = Opp
Adj
3.
Calculate xo
2nd
Tan ⁻¹
Tan
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Trigonometry
Now try
Exercise 4.2
MIA Page 205
Starter Questions
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1.
True or false 30  5 + 6  4 = 48
2. Write in scientific notation 0.0456
3. Identify the sides of the triangle.
xo
4. The subway train takes 6 mins to travel
between 2 stations 3 miles apart.
Show that it's average speed is 30mph.
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Angles & Triangles
Learning Intention
1. To use tan of the angle to
solve REAL LIFE problems.
Success Criteria
1. Write down tan ratio.
2. Use tan of an angle to solve
REAL LIFE problems.
16-Jul-15
Created by Mr. Lafferty Maths Dept.
Trigonometry
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Use the tan ratio to find the height h of the tree
to 2 decimal places.
tan 47o =
opp h
=
adj 8
tan 47o =
h
8
SOH CAH TOA
rod
h = 8 × tan 47o
h = 8.58m
16-Jul-15
47o
Compiled by Mr. Lafferty Maths Dept.
8m
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Trigonometry
SOH CAH TOA
Example 2
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.
16-Jul-15
What is the height of the plane ?
tan 6o =
h
15
c
h = 15 × tan 6o
h = 1.58km
Aeroplane
6o
Airport
Compiled by Mr. Lafferty Maths
Dept.
a = 15
Lennoxtown
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Trigonometry
Now try
Exercise 5.1
MIA Page 207
Starter Questions
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1.
Explain why 6 + 9  3 = 9 and not 5
2. Write in scientific notation 32.56
3. Identify the sides of the triangle.
xo
4. The train takes 10 mins to travel
between 2 stations 6miles apart.
Find the average speed of the train.
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Angles & Triangles
Learning Intention
1. To use tan of the angle to
find adjacent length.
Success Criteria
1. Write down tan ratio.
2. Use tan of an angle to solve
find adjacent length.
16-Jul-15
Created by Mr. Lafferty Maths Dept.
Trigonometry
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Use the tan ratio to calculate how far the ladder
is away from the building.
tan 45o =
opp 12
=
adj
d
12
d=
tan 45o
SOH CAH TOA
ladder
45o
d = 12m
16-Jul-15
Compiled by Mr. Lafferty Maths Dept.
dm
12m
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Trigonometry
Example 2
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?
tan 6o =
1.58
d
SOH CAH TOA
1.58
d=
tan 6o
d = 15 km
16-Jul-15
Aeroplane
a = 1.58 km
6o
Airport
Compiled by Mr. Lafferty Maths Dept.
Lennoxtown
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Trigonometry
Now try
Exercise 5.2
MIA Page 210
Starter Questions
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1.
3
3 5
+

4
5 7
2.
Explain why
y=
y = 6 when a = (-1) b = 2
(a - b)(b - a)(2a - b)
p
Q3. Given T = k .
v
Find k when T = 6 P = 18 and v = 9.
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Angles & Triangles
Learning Intention
1. To show how to find an
angle using tan ratio.
Success Criteria
1. Write down tan ratio.
2. Use tan ratio to find an angle.
16-Jul-15
Created by Mr. Lafferty Maths Dept.
Trigonometry
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Use the tan ratio to calculate the angle that the
support wire makes with the ground.
tan x o =
opp 11
=
adj
4
SOH CAH TOA
 11 
x = tan  
4
o
-1
11m
x o = 70o
16-Jul-15
Compiled by Mr. Lafferty Maths Dept.
xo
4m
Trigonometry
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Use the tan ratio to find the angle of take-off.
tan x o =
SOH CAH TOA
opp
88
=
adj 500
tan xo = 0.176
o
-1
o
x = tan (0.176) = 10
16-Jul-15
Compiled by Mr. Lafferty Maths Dept.
88m
xo
500 m
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Trigonometry
Now try
Exercise 6.1
MIA Page 211
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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)
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Angles & Triangles
Learning Intention
1. Definite the sine ratio and
show how to find an angle
using this ratio.
Success Criteria
1. Write down sine ratio.
2. Use sine ratio to find an
angle.
16-Jul-15
Created by Mr. Lafferty Maths Dept.
The Sine Ratio
Sin x° =
Opposite
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Trigonometry
x°
Opp
Hyp
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Find the
height h
Example
Trigonometry
h
Opp
Opp
Sin x° =
Hyp
Sin 34° =
11cm
34°
h
11
SOH CAH TOA
11 x Sin 34° = h
h = 11 x Sin 34° = 6.2cm (1 d.p.)
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Using Sin to calculate angles
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Example
Trigonometry
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Find the xo
6m
Opp
9m
Opp
Sin x° =
Hyp
6
Sin x° =
9
x°
SOH CAH TOA
Sin x° = 0.667 (3 d.p.)
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Trigonometry
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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
2nd
Sin ⁻¹
Sin
Followed by
Sin
Trigonometry
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Sin x° = 0.667 (3 d.p.)
Press
2nd
Enter 0.667
Sin ⁻¹
Sin
=
x = Sin ⁻¹0.667 = 41.8° (1 d.p.)
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Trigonometry
Now try
Exercise 7.1
MIA Page 212
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Starter Questions
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1. Explain why we can simply pick out Q1 , Q2 and Q3
then find 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 costs £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.
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Angles & Triangles
Learning Intention
1. To show how to use the
sine ratio to solve
Success Criteria
1. Write down sine ratio.
REAL-LIFE problems.
2. Use sine ratio to solve
REAL-LIFE problems.
16-Jul-15
Created by Mr. Lafferty Maths Dept.
Trigonometry
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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.
sin 70o =
opp
h
=
hyp 11.7
SOH CAH TOA
h = 11.7  sin70o
11.7m
h = 11m
16-Jul-15
Compiled by Mr. Lafferty Maths Dept.
70o
h
Trigonometry
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Use the sine ratio to find the angle of the ramp.
opp 10
sin x =
=
hyp 20
o
SOH CAH TOA
10
sin x =
20
o
 10 
x = sin   = 30o
 20 
o
16-Jul-15
-1
20 m
xo
Compiled by Mr. Lafferty Maths Dept.
10m
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Trigonometry
Now try
Exercise 7.2
MIA Page 214
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Starter Questions
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1. Fill in the ? marks.
2 x 2  9 x  7 = (2 x  ?)( x  ?)
2. Calculate A when
A=
2
w
4
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w = (-10)
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Angles & Triangles
Learning Intention
1. To show how to calculate
the hypotenuse using the
sine ratio.
Success Criteria
1. Write down sine ratio.
2. Use sine ratio to find the
hypotenuse.
16-Jul-15
Created by Mr. Lafferty Maths Dept.
Example
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Trigonometry
SOH CAH TOA
Opp
Sin x° =
Hyp
Sin 72° =
r=
5
r
5
sin 72o
r = 5.3 km
A road AB is right angled at B.
The road BC is 5 km.
Calculate the length of
the new road AC.
B
5km
C
72°
r
A
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Trigonometry
Now try
Exercise 8.1
MIA Page 215
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Starter Questions
1. Explain why we can simply pick out Q1 and Q3
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 discount of 80%
3. A lorry is travelling at 50mph.
It has travelled 75 miles.
Show that the time taken is 1hr 30 mins.
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Angles & Triangles
Learning Intention
1. Definite the cosine ratio
and show how to find an
length or angle using this
ratio.
Success Criteria
1. Write down cosine ratio.
2. Use cosine ratio to find a
length or angle.
16-Jul-15
Created by Mr. Lafferty Maths Dept.
The Cosine Ratio
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Trigonometry
Cos x° =
x°
Adjacent
Adj
Hyp
Example
Trigonometry
Adj
Cos x° =
Hyp
b
Cos 40° =
35
b
40°
Opp
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Find the
adjacent
length b
35mm
SOH CAH TOA
35 x Cos 40° = b
b = 35 x Cos 40°= 26.8mm (1 d.p.)
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Using Cos to calculate angles
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Example
Trigonometry
34cm
Adj
Cos x° =
Hyp
Cos x° =
34
45
Opp
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Find the
angle xo
x°
45cm
SOH CAH TOA
Cos x° = 0.756 (3 d.p.)
x = Cos ⁻¹0.756 =41°
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Trigonometry
Now try
Exercise 9.1
MIA Page 216
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Starter Questions
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1. Calculate 104 x 100
putting your answer in standard form.
2. Is this triangle right angled ?
If yes, find the size of angle xo .
If no find the area of the triangle.
6
xo
10
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8
The Three Ratios
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adjacent
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opposite
Sine
Tangent
Cosine
hypotenuse
adjacent
Sine
adjacent
Cosine
opposite
Cosine
Tangent
Sine
hypotenuse
opposite
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Sine
hypotenuse
Trigonometry
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Sin x° =
Opp
Hyp
Cos x° =
Adj
Hyp
O
A
S HC H
Tan x° =
O
T A
Opp
Adj
Trigonometry
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Process
1.
Write down
SOH CAH TOA
2.
3.
Identify what you want to find
what you know
Copy this!
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Trigonometry
Past Paper Type Questions
SOH CAH TOA
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Trigonometry
Past Paper Type Questions
SOH CAH TOA
(4 marks)
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Trigonometry
Past Paper Type Questions
SOH CAH TOA
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Trigonometry
Past Paper Type Questions
SOH CAH TOA
4 marks
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Trigonometry
Past Paper Type Questions
SOH CAH TOA
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Trigonometry
Past Paper Type Questions
SOH CAH TOA
(4marks)
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Trigonometry
Past Paper Type Questions
SOH CAH TOA
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Trigonometry
Past Paper Type Questions
SOH CAH TOA
(4marks)
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Trigonometry
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Trigonometry
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Trigonometry
Now try
Exercise 10.1 & 10.2
MIA Page 218