4.1 Radian and Degree measure Changing Degrees to Radians Linear speed Angular speed.
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Transcript of 4.1 Radian and Degree measure Changing Degrees to Radians Linear speed Angular speed.
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4.1 Radian and 4.1 Radian and Degree measure Degree measure
Changing Degrees to RadiansChanging Degrees to Radians
Linear speedLinear speed
Angular speedAngular speed
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Definition of an angle
An angle is made from two rays with a common initial point.
In standard position the initial side is on the x axis
side
Initial
side
alTer min
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Positive angle vs. Negative angle
Positive angles are Counter clockwise C.C.W.
Negative angles are Clockwise C.W.
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Angles with the same initial side and terminal side are coterminal.
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The measure of an angle is from initial side to terminal side
Vertex at the origin (Center)
r
r
Angle
Central
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Definition of a Radian
Radian is the measure of the arc of a unit circle.
Unit circle is a circle with a radius of 1.
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The quadrants in terms of Radians
What is the circumference of a circle with radius 1?
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The quadrants in terms of Radians
What is the circumference of a circle with radius 1? 2
1
20
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The quadrants in terms of Radians
The circumference can be cut into parts.
1
20
2
2
3
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The quadrants in terms of Radians
The circumference can be cut into parts.
1
20
2
2
3
I
20
II
2
III
2
3
IV
2
2
3
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Find the Coterminal Angle
Since equals 0. it can be added or subtracted from any angle to find a coterminal angle.
Given
2
4
3
4
52
4
3
4
112
4
3
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Complementary Angles – two angles are complementary
if their sum is 90 degrees or .2
Supplementary Angles have a sum of 180 degrees or .
Find the complementary and supplementary angles for
5
2
5
2
2
1010
4
10
5
5
25
3
5
2
5
5
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Radian vs. Degree measurements
360º =
180º =
So or
2
rad180
1
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Radian vs. Degree measurements
360º =
180º =
So or
To convert Degrees into Radians multiply by
To convert Radians into Degrees multiply by
2
rad180
1
180
1 rad
180
180
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Conversions: Radians Degrees
To convert degrees to radians, multiply by 180
rad
To convert radians to degrees, multiply by
rad
180
Converting an angle from to decimal form. "'SMD
"29'15152
3600
29
60
15152
25806.152
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Change 140º to RadiansChange to degrees
Use degree to rads.
Use rads to degrees
3
7
180
443460953.29
7
180
140
180*140
180
4203
1260180*
3
7
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How to use radian to find Arc length
The geometry way was to find the circumference of the circle and multiply by the fraction. Central angle
360º
In degrees Are length called S would be
rS 2
360
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How to use radian to find Arc length
In degrees Are length called S would be
In radian the equation is
rS 2
360
rS
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r = 9, θ = 215º Changing to rads
Are length S
r
36
43
180215
936
43
S
772.334
43
S
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Linear speed and Angular speed
Linear speed is
Angular speed is
Assuming “constant speed”
t
S
time
lengtharc
ttime
angleCentral
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Linear and Angular Speeds
Consider a particle moving at a constant speed along a circular arc ofradius r. If s is the length of the arc traveled in time t, then the linear speed v of the particle is
arc length
times
tLinear speed v
Moreover, if
is the angle (in radian measure) corresponding to the arc
length s, then the angular speed
(the lowercase Greek letter omega)
of the particle is
Angular speed
central angle
time
t
A relationship between linear speed and angular speed is
v r
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Area of a Sector of a Circle
A1
2r2
where is measured in radians
A sprinkler on a golf course fairway is set to spray water over a distanceof 70 feet and rotates through an angle of 120 degrees. Find the area of the fairway watered by the sprinkler.
12070 ft
120o = how many radians?
120 23radians
A1
270 2 2
3
49003
5131 ft 2
79-99 odd, 107
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Finding Linear Speed
The second hand of a clock is 10.2 cm long. Find the linear speed of theTip of the second hand as it passes around the clock face.
arc length
times
tLinear speed v
Arc length s
2r
2 (10.2)20.4 cm
,v s
t20.4 cm60 sec
=1.068 cm/sec