Transcript COMPLEX NUMBERS - science

Angular measurement
Objectives
Be able to define the radian.
Be able to convert angles from degrees into radians and
vice versa.
Outcomes
You MUST ALL be able to define the radian AND be able
to convert degrees into radians and vice-versa.
MOST of you SHOULD Be able to understand the
reasons for using radians AND be able to solve problems
involving a mixture of degrees and radians.
SOME of you COULD be able to work out arc length.
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
If the arc is 2r, the angle is 2 radians.
r
O
r
2c
2r
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
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
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
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
 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
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 
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
Extension
• Arc Length
Arc Length
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
Examples
1. Find the arc length, l, 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.
Examples
2. Find the arc length, l, 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


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
l  rθ
Exercises
1. Find the arc length, l,
of the sector shown.
O
2c
4 cm
l
2. Find the arc length, l, of the sector of a circle
of radius 8 cm. and sector angle 120. Give
exact answers in terms of

.
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.
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 
