Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be...
-
Upload
jeremy-evans -
Category
Documents
-
view
215 -
download
1
Transcript of Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be...
![Page 1: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/1.jpg)
Radians, Arc Length and Radians, Arc Length and Sector AreaSector Area
![Page 2: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/2.jpg)
Radians, Arc Length and Sector Area
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
![Page 3: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/3.jpg)
Radians, Arc Length and Sector Area
r
O
2r
r
2c
If the arc is 2r, the angle is 2 radians.
Radians
![Page 4: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/4.jpg)
Radians, Arc Length and Sector Area
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
![Page 5: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/5.jpg)
Radians, Arc Length and Sector Area
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
![Page 6: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/6.jpg)
Radians, Arc Length and Sector Area
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
![Page 7: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/7.jpg)
Radians, Arc Length and Sector Area
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
![Page 8: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/8.jpg)
Radians, Arc Length and Sector Area
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
![Page 9: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/9.jpg)
Radians, Arc Length and Sector Area
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
![Page 10: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/10.jpg)
Radians, Arc Length and Sector Area
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
![Page 11: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/11.jpg)
Radians, Arc Length and Sector Area
Arc Length and Sector Area
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
![Page 12: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/12.jpg)
Radians, Arc Length and Sector Area
O
r
r
θ
θrA 2
2
1
Also, the sector area A is the same fraction of the area of the circle.
A
2
Acircle area
2
2rθA
Arc Length and Sector Area
![Page 13: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/13.jpg)
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: where is in radians
θrl θ
cm.14)2)(7( ll
θrA 221 .cm249)2()7(
2
1 2 AA
![Page 14: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/14.jpg)
Radians, Arc Length and Sector Area
2. Find the arc length, l, and area, A, 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
θrA 221 .cm2
12
125
6
5)5(
2
1 2
AA
Examples
![Page 15: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/15.jpg)
Radians, Arc Length and Sector Area
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
θrA 221
For a sector of angle radians of a circle of radius r,
θ
• the arc length, l, is given by
• the sector area, A, is given by
SUMMARY
![Page 16: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/16.jpg)
Radians, Arc Length and Sector Area
1. Find the arc length, l, and area, A, of the sector shown. O
4 cm
A
c2
l
2. Find the arc length, l, and area, A, of the
sector of a circle of radius 8 cm. and sector
angle . Give exact answers in terms of
.
120
Exercises
![Page 17: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/17.jpg)
Radians, Arc Length and Sector Area
1. Solution:
θrl cm.8)2)(4( l
θrA 221 .cm216)2()4( 2
21 A
O4 cm
A
c2
l
Exercises
![Page 18: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/18.jpg)
Radians, Arc Length and Sector Area
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
![Page 19: Radians, Arc Length and Sector Area. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649f3b5503460f94c596b0/html5/thumbnails/19.jpg)
Radians, Arc Length and Sector Area