Transcript 6)_C2_Radian_Measure_and_its_Applications

Introduction
• This Chapter focuses on using ‘Radians’ when
answering questions involving circles
• Radians are an alternative to degrees
• Radians are quicker to use than degrees (when
you get used to them)
• They also allow extra calculations which would
be much more difficult to do using degrees
instead…
Radian measure and its
Applications
You can measure angles in Radians
Radians are an alternative to degrees.
Some calculations involving circles are
easier when Radians are used, as
opposed to degrees.
r
A
O 1c
r
r
B
‘If arc AB has length r, then angle AOB
is 1 radian (1c or 1 rad)’
Arc Length
r  1c
Multiply by 2π
c
Arc Length 2 r  2
360  2 c
180  
180
 1c

c
2πr is the
circumference
÷2
÷π
6A
Radian measure and its
Applications
You can measure angles in Radians
Convert the following
angle to degrees
You need to be able to convert between
degrees and radians.
Radians  Degrees
c
1

180

Multiply
by 180/π
Multiply by
180/
π
Top x Top,
Bottom x Bottom
Cancel out π
Work out
the sum
7
rad
8
7 180

8

1260
8
1260
8
157.5
6A
Radian measure and its
Applications
You can measure angles in Radians
Convert the following
angle to degrees
You need to be able to convert between
degrees and radians.
Radians  Degrees
c
1

180

Multiply
by 180/π
Multiply by
180/
π
Top x Top,
Bottom x Bottom
Cancel out π
Work out
the sum
4
rad
15
4 180

15 
720
15
720
15
48
6A
Radian measure and its
Applications
You can measure angles in Radians
Convert the following
angle to radians
You need to be able to convert between
degrees and radians.
Degrees  Radians
c
1

180

Divide
by 180/π
Multiply
by π/180
Multiply by
π/
180
Only multiply the
top here
Simplify
150
150 

180
150
180
5
rad
6
6A
Radian measure and its
Applications
You can measure angles in Radians
Convert the following
angle to radians
You need to be able to convert between
degrees and radians.
Degrees  Radians
c
1

180

Divide
by 180/π
Multiply
by π/180
Multiply by
π/
180
Only multiply the
top here
Simplify
110
110 

180
110
180
11
rad
18
6A
Radian measure and its
Applications
Finding the length of an arc is
easier when you use radians
Length of Arc
Circumference
=
Angle at Centre
Total Angle at Centre

=
2
2 r
r
θ
l
r
l
l
r
=
l
= r
Multiply
by 2π

Multiply
by r
(The angle must be in radians!)
6B
Radian measure and its
Applications
Finding the length of an arc is
easier when you use radians
Find the length of the arc of a
circle of radius 5.2cm. The arc
subtends an angle of 0.8c at the
centre of the circle.
l
= r
l
= 5.2  0.8
l
=
4.16cm
6B
Radian measure and its
Applications
Finding the length of an arc is
easier when you use radians
Arc AB of a circle, with centre O
and radius r, subtends an angle of
θ radians at O. The Perimeter of
sector AOB is P cm. Express r in
terms of θ.
A
r
O
Length AB = rθ
Factorise
Divide by
(θ + 2)
P  r  2r
P  r (  2)
P
 r
(  2)
rθ
θ
r
B
6B
Radian measure and its
Applications
Finding the length of an arc is
easier when you use radians
The border of a garden pond consists
of a straight edge AB of length 2.4m,
and a curved part C, as shown in the
diagram below. The curved part is an
arc of a circle, centre O and radius 2m.
Find the length of C.
C
opp
hyp
1.2
sin x 
2
sin x 
Inverse sine
Double for
angle AOB
sin x  0.6
Calculator
in Radians
(H)
2m
A
x  0.6435rad
1.2m
(O)
O
x
B
2 x  1.287rad
Angle θ = 2π – 1.287
Angle θ = 4.996 rad
4.996
θ c
2m O
1.287c
A
2.4m
2m
B
(We need to work out angle θ)
l  r
l  2  4.996
l  9.99m
6B
Radian measure and its
Applications
The Area of a Sector and Segment
can be worked out using Radians
Area of Sector
Angle at Centre
=
Total Area
Total Angle
A
O
θ
Multiply by π
X
X
 r2
=

2
X
r2
=

2
Multiply by r2
B
X
=
X
=
 r2
2
1 2
r
2
This is the formula’s
usual form
6C
Radian measure and its
Applications
The Area of a Sector and Segment
can be worked out using Radians
In the diagram, the area of the minor
sector AOB is 28.9cm2. Given that
angle AOB is 0.8 rad, calculate the
value of r.
A
Put the
numbers in
½ x 0.8 =
0.4
c
0.8
B
28.9 
1 2
r
2
1 2
r (0.8)
2
28.9  0.4r 2
Divide by
0.4
r cm
O
A
Square root
72.25  r 2
8.5cm  r
6C
Radian measure and its
Applications
The Area of a Sector and Segment
can be worked out using Radians
A plot of land is in the shape of a
sector of a circle of radius 55m. The
length of fencing that is needed to
enclose the land is 176m. Calculate the
area of the plot of land.
The length of the arc
must be 66m (adds up
to 176 total)
Put the
numbers in
Divide by 55
A
55m
66m
1.2θc
O
55m
B
(We need to work out the angle first)
Put the
numbers in
l  r
66  55
1.2c  
1 2
A r 
2
1
A   552  
2
A  1815 m2
6C
Radian measure and its
Applications
The Area of a Sector and Segment
can be worked out using Radians
Area of a Segment
Area of Sector AOB – Area of Triangle AOB
Area of Sector AOB
You can also work out the area of a
segment using radians.
A
1 2
r
2
Area of Triangle AOB
a=b=r
C=θ
O
r
A
θ
r
1
ab sin C
2
1
A  r 2 sin 
2
A
Area of the Segment
B
Factorise
1 2 1 2
r   r sin 
2
2
1 2
r (  sin  )
2
6C
Radian measure and its
Applications
The Area of a Sector and Segment
can be worked out using Radians
Calculate the Area of the segment shown in
the diagram below.
Substitute the
numbers in
Work the
parts out
1 2
r (  sin  )
2
1 2

2.5   sin 
2
3
3
3.125  0.1811...
Only round the
final answer
O
π
3
2.5cm
0.57 cm2
6C
Radian measure and its
Applications
The Area of a Sector and Segment
can be worked out using Radians
Area of the shaded segment
In the diagram AB is the diameter of a
circle of radius r cm, and angle BOC = θ
radians. Given that the Area of triangle
AOC is 3 three times that of the shaded
segment, show that 3θ – 4sinθ = 0.
Area of triangle AOC
1 2
r   sin  
2
a=b=r
Angle = π-θ
1
ab sin C
2
1 2
r sin(   )
2
Remember, sin x = sin (180 – x)
C
A
0
θ
1 2
r sin 
2
B
AOC = 3 x
shaded segment
Cancel out 1/2r2
Multiply out the
brackets
Subtract sinθ
1 2
1
r sin   3  r 2   sin  
2
2
sin   3  sin  
sin   3  3sin 
0  3  4sin 
6C
Summary
• We have learnt how to change from
degrees to radians
• We have seen how to do calculations to
work out the length of an arc
• We have also seen formulae for the
Area or a sector and segment