Angular measurementAngular measurement
ObjectivesBe able to define the radian.Be able to convert angles from degrees into radians and vice versa.
OutcomesOutcomes
You MUST ALLMUST ALL be able to define the radian AND be able to convert degrees into radians and vice-versa.
MOSTMOST of you SHOULDSHOULD Be able to understand the reasons for using radians AND be able to solve problems involving a mixture of degrees and radians.
SOMESOME of you COULDCOULD be able to work out arc length.
Radians
Radians are units for measuring angles.They can be used instead of degrees.
r
O
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.
r
r
x = 1 radian
x
= 1 rad. or 1c
r
O
2r
r
2c
If the arc is 2r, the angle is 2 radians.
Radians
O
If the arc is 3r, the angle is 3 radians.
r3r
r
3c
If the arc is 2r, the angle is 2 radians.
Radians
O
If the arc is 3r, the angle is 3 radians.
c143
If the arc is 2r, the angle is 2 radians.
r
r
If the arc is r, the angle is radians.
143 143
r143
Radians
O
If the arc is 3r, the angle is 3 radians.
r
r
If the arc is 2r, the angle is 2 radians.
If the arc is r, the angle is radians.
143 143
If the arc is r, the angle is radians.
rc
Radians
If the arc is r, the angle is radians.
O
r
r
rc
But, r is half the circumference of the circle so the angle is
180
180 radians Hence,
Radians
We sometimes say the angle at the centre is subtended by the arc.
180 radians
Hence,
180
357
radian 1
r
O
r
rx
x = 1 radian357
Radians
Radians
SUMMARY
• One radian is the size of the angle subtended by the arc of a circle equal to the radius
180 radians •
• 1 radian 357
Exercises
1. Write down the equivalent number of degrees for the following number of radians:
Ans:
(a) (b) (c) (d)2
3
26
(a) (b) (c) (d)60 45 120 30
2. Write down, as a fraction of , the number of radians equal to the following:
(a) (b) (c) (d)6090 360 30
(a) (b) (c) (d)3
6
32
4
Ans:
It is very useful to memorize these conversions
Extension
• Arc Length
Arc Length
Let the arc length be l .
O
r
r
l
rl 22
Consider a sector of a circle with angle .
θ
Then, whatever fraction is of the total angle at O, . . .
θ
θrl
2
θ. . . l is the same fraction of the circumference. So,
( In the diagram this is about one-third.)
2
l circumference
2
lcircumference
Examples
1. Find the arc length, l, of the sector of a circle of radius 7 cm. and sector angle 2 radians.
Solution: where is in radians
θrl θ
cm.14)2)(7( ll
2. Find the arc length, l, of the sector of a circle of radius 5 cm. and sector angle . Give exact answers in terms of .
150
Solution: where is in radians
θrl θ180 rads.
630
rads
. 6
5150
rads.
So, cm.6
25
6
55
llrθl
Examples
Radians• An arc of a circle equal in length to the
radius subtends an angle equal to 1 radian. 180 radians •
• 1 radian 357
θrl
For a sector of angle radians of a circle of radius r,
θ
• the arc length, l, is given by
SUMMARY
1. Find the arc length, l, of the sector shown. O
4 cmc2
l
2. Find the arc length, l, of the sector of a
circle of radius 8 cm. and sector angle .
Give exact answers in terms of .
120
Exercises
1. Solution:
θrl cm.8)2)(4( l
θrA 221 .cm216)2()4( 2
21 A
O4 cm
A
c2
l
Exercises
2. Solution:
180 rads.
360
rads
.
3
2120
rads.
So, cm.3
16
3
28
llrθl
θrA 221 .cm2
3
64
3
2)8(
2
1 2
AA
O8 cm
A
120
l
where is in radians
θrl
Exercises
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