Transcript 5.1 Radian and Degree Measure Measuring Angles

Angles and Arcs in the Unit Circle
1
In this section, we will study the following
topics:
Terminology used to describe angles
 Degree measure of an angle
 Radian measure of an angle
 Converting between radian and degree
measure
 Finding coterminal angles

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Angles
Trigonometry: measurement of triangles
Section 4.1, Figure 4.1, Terminal and
Angle Measure
Initial Side of an Angle , pg. 248
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Digital Figures, 4–2
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Standard Position
Section 4.1, Figure 4.2, Standard
Position of an Angle, pg. 248
Vertex at origin
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The initial side of an angle
in standard position is always located
on the positive x-axis.
Digital Figures, 4–3
4
Positive and Section
negative4.1,
angles
Figure 4.3, Positive and
Negative Angles, pg. 248
When sketching angles,
always use an arrow to
show direction.
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Digital Figures, 4–4
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Measuring Angles
The measure of an angle is determined by the amount of
rotation from the initial side to the terminal side.
There are two common ways to measure angles, in degrees
and in radians.
We’ll start with degrees, denoted by the symbol º.
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One degree (1º) is equivalent to a rotation of
360
revolution.
of one
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Section 4.1, Figure 4.13, Common Degree
Measuring Angles
Measures on the Unit Circle, pg. 251
1
360
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Digital Figures, 4–9
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Classifying Angles
Angles are often classified according to the
quadrant in which their terminal sides lie.
Ex1: Name the quadrant in which each angle
lies.
50º
Quadrant 1
208º
Quadrant 3
II
I
-75º
Quadrant 4
III
IV
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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, 0º, 90º, 180º, 270º, 360º, … are
quadrantal angles.
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Coterminal Angles
4.1, Figure
4.4,
Coterminal
Angles thatSection
have the same
initial and
terminal
sides are
coterminal.
Angles, pg. 248
Angles  and  are coterminal.
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Digital Figures, 4–5
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Example of Finding Coterminal Angles
You can find an angle that is coterminal to a given angle  by
adding or subtracting multiples of 360º.
Ex 2:
Find one positive and one negative angle that are
coterminal to 112º.
For a positive coterminal angle, add 360º : 112º + 360º = 472º
For a negative coterminal angle, subtract 360º: 112º - 360º = -248º
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Ex. 3: Find one positive and one negative angle
that is coterminal with the angle  = 30 o in
standard position.
Ex. 4: Find one positive and one negative angle
that is coterminal with the angle  = 272 in
standard position.
Radian Measure
A second way to measure angles is in radians.
Definition of Radian:
4.1, Figure
Illustration
One radian isSection
the measure
of a 4.5,
central
angle ofthat intercepts
pg. r249
arc s equal in lengthArc
to Length,
the radius
of the circle.
In general,
s

r
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Digital Figures, 4–6
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Radian Measure
2 radians corresponds to 360
2  6.28
 radians corresponds to 180

  3.14

2
2
radians corresponds
904.6, Illustration of
Section 4.1,to
Figure
Six Radian Lengths, pg. 249
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 1.57
Digital Figures, 4–7
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Section 4.1, Figure 4.7, Common
Radian Measure
Radian Angles, pg. 249
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Conversions Between Degrees and Radians
1.
To convert degrees to radians, multiply degrees by
2.
To convert radians to degrees, multiply radians by

180
180

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a)
60
b)
30
c)
-54
d)
-118
e)
45
a)

b) 6

c)
2
11
d) 
18
e) 
9

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Ex 8: Find one positive and one negative angle that
is coterminal with the angle  = 7 in standard
position.
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Degree and Radian Form of “Special” Angles
90 ° 
 120 °
60 ° 
 135 °
45 ° 
 150 °
30 ° 

0° 
 180 °
360 ° 
 210 °
330 ° 
 225 °
315 ° 
 240 °
300 ° 
270 ° 
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Convert from degrees to radians.
1. 54
2. -300
Convert from radians to degrees.
3.
11
4. 3
13

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Find one postive angle and one negative angle
in standard position that are coterminal with
the given angle.
5. 135
6.
11
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