Transcript Radians Introduction

RADIANS
Definition
An arc of length r subtends an
angle of one radian at the
centre of a circle of radius r
r
r
Ø=1radian
r
Proof
r
r
How do you calculate the length of an arc?
Ø=1radian
r
x 360 º
r = ø x 2πr
360º
r = 1 radian x 2πr
360 º
360r = 1 radian x 2πr
÷r
360 º = 1 radian x 2π
÷2
180 º = 1 radian x π
÷π
180 = 1 radian
π
so 1 radian is approximately…?
Or 180 º = π radians
Converting between angles and radians
Degrees = radians x 180
π
Radians = degrees x π
180
So if ø is measured in radians
Then ø radians = ø x π
180
How many different angles can you write
as radians?
90º = π radians
2
180º = π radians
r
r
Arc Length
Ø=1radian
r
Arc length = ø x 2πr
360º
Angle in
degrees
Arc length = 2πrø
360º
Factorise r
Arc length = r 2πø
360º
Divide by 2
Arc length = r πø
180º
Arc length = rø
Angle
in
radians
r
r
Area of Sector
Ø=1radian
r
Sector area = ø x πr2
360º
Angle in
degrees
Sector area = πr2ø
360º
Factorise r2
Sector area = r2 πø
360º
Factorise out ½
Sector area = ½r2 πø
180º
Sector area = ½r2ø
Angle
in
radians
Examples
Convert 50° into radians
50° =
50° x π rad
180
50° =
0.87 rad
Examples
Convert 2.7 radians into degrees
2.7 rad =
2.7 x 180
π
2.7 rad = 154.7 °
degrees
Examples
Convert 40° into radians
40° = 40 x π
180
40° = 40π
180
40° = 2π radians
9
10cm
1.2radians
Calculate the arc
length and sector area
Arc length = rө
Arc length = 10 x 1.2
Arc length = 12cm
Sector area = ½r2ө
Area = ½ x 100 x 1.2
Area = 60cm2