Transcript COMPLEX NUMBERS

“Teach A Level Maths”
Vol. 1: AS Core Modules
40: Radians, Arc Length and
Sector Area
© Christine Crisp
Radians, Arc Length and Sector Area
Module C2
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Radians, Arc Length and Sector Area
Radians
Radians are units for measuring angles.
They can be used instead of degrees.
1 radian is the size of the
angle formed at the centre of
a circle by 2 radii which join
the ends of an arc equal in
length to the radius.
O
x
r
r
x = 1 radian
= 1 rad. or 1
r
c
Radians, Arc Length and Sector Area
Radians
If the arc is 2r, the angle is 2 radians.
r
O
r
2c
2r
Radians, Arc Length and Sector Area
Radians
If the arc is 2r, the angle is 2 radians.
If the arc is 3r, the angle is 3 radians.
r
O 3c
r
3r
Radians, Arc Length and Sector Area
Radians
If the arc is 2r, the angle is 2 radians.
If the arc is 3r, the angle is 3 radians.
If the arc is 3 14 r, the angle is 3 14 radians.
r
O 3  14 c
r
3 14 r
Radians, Arc Length and Sector Area
Radians
If the arc is 2r, the angle is 2 radians.
If the arc is 3r, the angle is 3 radians.
If the arc is 3 14 r, the angle is 3 14 radians.
r
O

c
r
r
If the arc is
 r, the angle is 
radians.
Radians, Arc Length and Sector Area
Radians
If the arc is
 r, the angle is 
radians.
r
O
c
r
r
But,  r is half the circumference of the circle
so the angle is 180
Hence,
 radians  180
Radians, Arc Length and Sector Area
Radians
 radians  180
Hence,

180
1 radian 

 57 3 
O
x
x = 1 radian
r
r
 57 3 
We sometimes say the angle at the centre
is subtended by the arc.
r
Radians, Arc Length and Sector Area
SUMMARY
 Radians
• One radian is the size of the angle subtended
by the arc of a circle equal to the radius
•
 radians  180
•
1 radian  57 3 
Radians, Arc Length and Sector Area
Exercises
1. Write down the equivalent number of degrees
for the following number of radians:
(a)

(b)

(c)
2
3
2
(d)

6
Ans: (a) 90  (b) 60  (c) 360  (d) 30 
It is very useful to memorize these conversions
2. Write down, as a fraction of  , the number of
radians equal to the following:
(a) 60 
(b) 45 
(c) 120
(d) 30 
Ans:
(a)

3
(b) 
4
(c) 2
3
(d) 
6
Radians, Arc Length and Sector Area
Arc Length and Sector Area
Consider a sector of a circle with angle
Let the arc length be l .
Then, whatever fraction θ is of
the total angle at O, . . .
. . . l is the same fraction of the
circumference. So,
l
 
2
θ.
circumference
( In the diagram this is about one-third.)
 l    circumference
2
 l    2r

l

r
θ
2
2
O
r
r
θ
l
Radians, Arc Length and Sector Area
Arc Length and Sector Area
Also, the sector area A is the same
fraction of the area of the circle.
 
circle area
2
A
θr
2
1 2
A r θ
2
 A 

2
r
O
r
θ A
Radians, Arc Length and Sector Area
Examples
1. Find the arc length, l, and area, A, of the sector
of a circle of radius 7 cm. and sector angle 2
radians.
Solution:
l  r θ where θ is in radians
 l  (7)(2)  l  14 cm.
A  12 r 2 θ
1
 A
(7) 2 ( 2)  A  49 cm 2 .
2
Radians, Arc Length and Sector Area
Examples
2. Find the arc length, l, and area, A, of the sector
of a circle of radius 5 cm. and sector angle 150 .
Give exact answers in terms of  .
Solution:
l  r θ where θ is in radians
 rads.
 5  rads.
 rads.  180  30 
 150 
6
6
5
25
So, l  rθ  l  5 
 l 
cm.
6
6
1
125
2  5 
2
1
( 5) 
cm 2 .
  A
A 2r θ  A 
2
12
 6 


Radians, Arc Length and Sector Area
SUMMARY
 Radians
• An arc of a circle equal in length to the
radius subtends an angle equal to 1 radian.
•
 radians  180
•
1 radian  57 3 
 For a sector of angle
of radius r,
θ
radians of a circle
•
the arc length, l, is given by
•
the sector area, A, is given by
l  rθ
A
1 r 2θ
2
Radians, Arc Length and Sector Area
Exercises
1. Find the arc length, l,
and area, A, of the
sector shown.
O
2c
4 cm
A
l
2. Find the arc length, l, and area, A, of the sector
of a circle of radius 8 cm. and sector angle 120 .
Give exact answers in terms of
.
Radians, Arc Length and Sector Area
Exercises
1. Solution:
O
2c
4 cm
A
l
 l  (4)(2)  8 cm.
l  rθ
A
1 r2
2
θ
 A  12 (4) 2 (2)  16 cm 2.
Radians, Arc Length and Sector Area
Exercises
2. Solution:
l  r where θ is in radians
 rads.  180
O

8 cm 120
 rads.
 60 
A
3
 2  rads.
 120 
l
3
2
16
So, l  rθ  l  8 
 l 
cm.
3
3
1
64
2  2 
2
1
( 8) 
cm 2 .
  A
A 2r θ  A 
2
3
 3 

Radians, Arc Length and Sector Area
Radians, Arc Length and Sector Area
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
Radians, Arc Length and Sector Area
SUMMARY
 Radians
• One radian is the size of the angle subtended
by the arc of a circle equal to the radius
•
 radians  180
•
1 radian  57 3 
O
x
r
r
r
Radians, Arc Length and Sector Area
Arc Length and Sector Area
Consider a sector of a circle with angle
Let the arc length be l .
Then, whatever fraction θ is of
the total angle at O, . . .
. . . l is the same fraction of the
circumference. So,
l
 
2
θ.
circumference
( In the diagram this is about one-third.)
 l    circumference
2
 l    2r

l

r
θ
2
2
O
r
r
θ
l
Radians, Arc Length and Sector Area
Arc Length and Sector Area
Also, the sector area A is the same
fraction of the area of the circle.
 
circle area
2
A
θr
2
1 2
A r θ
2
 A 

2
r
O
r
θ A
Radians, Arc Length and Sector Area
SUMMARY
 For a sector of angle
of radius r,
θ
radians of a circle
•
the arc length, l, is given by
•
the sector area, A, is given by
l  rθ
A
1 r 2θ
2