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