Radian MeasureRadian Measure

Angles can be measured 3 ways:Angles can be measured 3 ways: 1) Degrees 1) Degrees (360 parts to a rotation)(360 parts to a rotation)

used for triangle applicationsused for triangle applications

2) Radians2) Radians (2(2∏∏ parts to a rotation) parts to a rotation) used for circular applicationsused for circular applications

3) Grads3) Grads (100 parts to a rotation)(100 parts to a rotation) used by the militaryused by the military

Radian MeasureRadian Measure

Place an anglewith the vertexat the center of a circle.

Radian MeasureRadian Measure

An anglemeasured in radians

is the ratio of thearc length of the circle

to the radius of the circle.

radius

θ

What is a radian?What is a radian?

In science and engineering, In science and engineering, radiansradians are are much more convenient (and common) much more convenient (and common) than degrees. A than degrees. A radianradian is defined as the is defined as the angle between 2 radii (radiuses) of a circle angle between 2 radii (radiuses) of a circle where the arc between them has length of where the arc between them has length of one radius. one radius. Another way of putting it is: "a radian is Another way of putting it is: "a radian is the angle subtended by an arc of length the angle subtended by an arc of length rr (the radius)".(the radius)".One radian is about 57.3°.One radian is about 57.3°.

Radian MeasureRadian Measure

r

srad

r

srad

radius

θ

radius

lengtharcradiansinangle

rS We will use the formula:

where S is the arc length,

r is the radius, and

is the angle in radians

Find the length of the arc shown:Find the length of the arc shown:(round to the nearest tenth)(round to the nearest tenth)

)13(4

3S

rS

=30.6 ft

Find the central angle:Find the central angle:(round to the nearest tenth)(round to the nearest tenth)

60)(5.146

rS

=2.4 radians

Radian MeasureRadian Measure

Conversion Factor:

180

180

or

Sometimes it is necessary to convert radians to degrees and degrees to radians.

Radian MeasureRadian Measure

180

30

30° can be converted to radians by…

6

Radian MeasureRadian Measure

180

630

can be converted to degrees by…

6

Whiteboards Ready?Whiteboards Ready?

Radian MeasureRadian Measure

45°= _____________ radians

45180

Think…

4

Radian MeasureRadian Measure

60°= _____________ radians

3

Think… 60180

Radian MeasureRadian Measure

315°= _____________ radians

4

7Think…

180

315

Radian MeasureRadian Measure

120°= _____________ radians

3

2Think…

180

120

Radian MeasureRadian Measure

radians= _______ degrees

1354

3

180

4

3Think…

Radian MeasureRadian Measure

radians= _______ degrees

902

180

2Think…

Radian MeasureRadian Measure

radians= _______ degrees

270

2

3

180

2

3Think…

Radian MeasureRadian Measure

radians= _______ degrees

300

3

5

180

3

5Think…

How about this?How about this?Find the exact value of Find the exact value of sinsin

6

2

130sin

30180

6

Go to the Practice!!!Go to the Practice!!!

What is ?What is ?

Have fun.Have fun.

Mwahaha…..Mwahaha…..

b

b

tan

sin