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Angular measurementAngular measurement
ObjectivesBe able to define the radian.Be able to convert angles from degrees into radians and vice versa.
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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.
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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
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r
O
2r
r
2c
If the arc is 2r, the angle is 2 radians.
Radians
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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
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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
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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
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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
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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
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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
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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
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Extension
• Arc Length
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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
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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
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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
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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
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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
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1. Solution:
θrl cm.8)2)(4( l
θrA 221 .cm216)2()4( 2
21 A
O4 cm
A
c2
l
Exercises
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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