Radians

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Angles and
Their Measure
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formed by rotating a ray about its endpoint
(vertex)
Angle-
Ending position
Starting position
Initial side on positive x-axis and the vertex at
the origin
Standard Position
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Angles measured counterclockwise are given a
positive sign and angles measured clockwise are
given a negative sign.
Terminal Side
Positive Angle
This is a counterclockwise rotation.
Negative Angle
This is a clockwise rotation.
Initial Side
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Its Greek To Me!
It is customary to use small letters in the Greek
alphabet to symbolize angle measurement.
?
?
?
alpha
beta
gamma
?
?
?
theta
delta
phi
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Standard Position
Vertex at origin
The initial side of an angle in standard
position is always located on the positive
x-axis.
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We can use a coordinate system with angles by
putting the initial side along the positive
x-axis with the vertex at the origin.
Quadrant IIangle
Quadrant Iangle
Terminal Side
? positive
Initial Side
? negative
Quadrant IVangle
If the terminal side is along an axis it is
called a quadrantal angle.
We say the angle lies in whatever quadrant the
terminal side lies in.
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We will be using two different units of measure
when talking about angles Degrees and Radians
If we start with the initial side and go all of
the way around in a counterclockwise direction we
have 360 degrees
? 360
? 90
If we went 1/4 of the way in a clockwise
direction the angle would measure -90
You are probably already familiar with a right
angle that measures 1/4 of the way around or
90 ¼(360 ) 90
? - 90
Lets talk about degrees first. You are probably
already somewhat familiar with degrees.
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What is the measure of this angle?
You could measure in the positive direction and
go around another rotation which would be another
360
? - 360 45
? - 315
? 45
You could measure in the positive direction
? 360 45 405
You could measure in the negative direction
There are many ways to express the given angle.
Whichever way you express it, it is still a
Quadrant I angle since the terminal side is in
Quadrant I.
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Measuring Angles
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Classifying Angles
Standard position angles that have their terminal
side on one of the axes are called quadrantal
angles. For example, 0, 90, 180, 270, 360,
are quadrantal angles.
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Radian Measure
A second way to measure angles is in
radians. Definition of Radian One radian is the
measure of a central angle ? where its arc s is
equal to length of the radius r of the circle.
In general,
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• Radian is the measure of the arc of a unit
  circle.
• Unit circle is a circle with a radius of 1.

s
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1 Radian measure of central angle, ?, that
intercepts the arc that has the same length as
the radius of the circle
Arc length s radius when ? 1 radian
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Calculate the number of radians in one full
circle
C
? 3.14
0, 2? 0, 6.28
0
Therefore, we can say that 1 full revolution 2?
radians.
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Radian Measure
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Radian Measure
360
180
90
60
45
30
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Convert to radians
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  To convert from degrees to radians, multiply by
  To convert from radians to degrees, multiply
by
Convert to radians
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  To convert from degrees to radians, multiply by
  To convert from radians to degrees, multiply
by
Convert to degrees
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  To convert from degrees to radians, multiply by
  To convert from radians to degrees, multiply
by
Convert to degrees
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  To convert from degrees to radians, multiply by
  To convert from radians to degrees, multiply
by
Convert to degrees
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