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Trigonometric Functions:
Right Triangle Approach
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6.1
Angle Measure
Copyright © Cengage Learning. All rights reserved.
Objectives
► Angle Measure
► Angles in Standard Position
► Length of a Circular Arc
► Area of a Circular Sector
► Circular Motion
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Angle Measure
An angle AOB consists of two rays R1 and R2 with a
common vertex O (see Figure 1).
We often interpret an angle as a rotation of the ray
R1 onto R2.
Negative angle
Positive angle
Figure 1
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Angle Measure
In this case, R1 is called the initial side, and R2 is called the terminal
side of the angle. An angle is said to be in standard position if its vertex
is at the origin of the circle and its initial side is along the positive xaxis.
If the rotation is counterclockwise, the angle is considered positive,
and if the rotation is clockwise, the angle is considered negative.
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Angle Measure
The measure of an angle is the amount of rotation about
the vertex required to move R1 onto R2.
Intuitively, this is how much the angle “opens.” One unit of
measurement for angles is the degree.
An angle of measure 1 degree is formed by rotating the
initial side
of a complete revolution.
In calculus and other branches of mathematics, a more
natural method of measuring angles is used—radian
measure.
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WHAT IS RADIAN MEASURE?
In some applications, it is easier to
associate the measure of an angle
with the length of an arc of a
circle centered at the vertex of
the angle.
This is the idea of radian measure,
a radian is the measure of an
angle that, when drawn as a
central angle of a circle,
intercepts an arc whose length is
equal to the length of the radius
of the circle.
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Angle Measure
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Figure 2
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Angle Measure
Now since the circumference of the circle of radius 1 is 2
a complete revolution of a circle has measure 2 rad, as
does a straight angle have a measure of  rad, and a right
angle the measure  /2 rad. Other angle measures are
determined by the length of the arc intersected by the
central angle:
An angle that is subtended by an arc of length 2 along the
unit circle has radian measure 2 (see Figure 3).
Radian measure
Figure 3
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Angle Measure
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Example 1 – Converting Between Radians and Degrees
(a) Express 60 in radians. (b) Express
rad in degrees.
Solution:
The relationship between degrees and radians gives
(a) 60
(b)
= 30
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Angle Measure
A note on terminology: We often use a phrase such as
“a 30 angle” to mean an angle whose measure is 30.
Also, for an angle , we write  = 30 or  =  /6 to mean
the measure of  is 30 or  /6 rad.
When no unit is given, the angle is assumed to be
measured in radians.
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Angles in Standard Position
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Angles in Standard Position
An angle is in standard position if it is drawn in the
xy-plane with its vertex at the origin and its initial side on
the positive x-axis.
Figure 5 gives examples of angles in standard position.
(a)
(b)
(c)
(d)
Angles in standard position
Figure 5
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Angles in Standard Position
Two angles in standard position are coterminal if their
sides coincide.
In Figure 5 the angles in (a) and (c) are coterminal.
(a)
(b)
(c)
(d)
Angles in standard position
Figure 5
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Example 2 – Coterminal Angles
(a) Find angles that are coterminal with the angle  = 30 in
standard position.
(b) Find angles that are coterminal with the angle  =
standard position.
in
Solution:
(a) To find positive angles that are coterminal with , we
add any multiple of 360.
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Example 2 – Solution
cont’d
Thus
30° + 360° = 390°
and
30° + 720° = 750°
are coterminal with  = 30. To find negative angles that
are coterminal with , we subtract any multiple of 360°.
Thus
30° – 360° = –330°
690°
and
30° – 720° = –
are coterminal with .
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Example 2 – Solution
cont’d
See Figure 6.
Figure 6
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Example 2 – Solution
cont’d
(b) To find positive angles that are coterminal with , we
add any multiple of 2.
Thus
and
are coterminal with  =  /3.
To find negative angles that are coterminal with , we
subtract any multiple of 2.
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Example 2 – Solution
cont’d
Thus
and
are coterminal with  =  /3. To find negative angles that
are coterminal with , we subtract any multiple of 2.
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Example 2 – Solution
cont’d
Thus
and
are coterminal with . See Figure 7.
Figure 7
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Length of a Circular Arc
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LENGTH OF AN ARC
An arc of a circle is a "portion" of
the circumference of the circle.
The length of an arc is simply
the length of its "portion" of the
circumference.
The length of an arc (or arc
length) is traditionally symbolized
by s.
In the diagram at the right, it can
be said that
subtends angle
and that the measure of θ is the
part of the circle determined by
the arc rotation.
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Length of a Circular Arc
Recall the formula for length of an arc from
Geometry:
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Length of a Circular Arc
Now using Radian measure, we obtain the formula:
Length of Arc = θ * 2π r
2π
Solving for , we get the important formula
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Length of a Circular Arc
This formula allows us to define radian measure using a
circle of any radius r: The radian measure of an angle  is
s/r, where s is the length of the circular arc that subtends 
in a circle of radius r (see Figure 10).
The radian measure of  is the number of “radiuses” that
can fit in the arc that subtends  ; hence the term radian.
Figure 10
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Example 4 – Arc Length and Angle Measure
(a) Find the length of an arc of a circle with radius 10 m that
subtends a central angle of 30.
(b) A central angle  in a circle of radius 4 m is subtended
by an arc of length 6 m. Find the measure of  in
radians.
Solution:
(a) From Example 1(b) we see that 30 =  /6 rad. So the
length of the arc is
s = r =
=
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Example 4 – Solution
cont’d
(b) By the formula  = s/r, we have
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Area of a Circular Sector
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Area of a Circular Sector
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Example 5 – Area of a Sector
Find the area of a sector of a circle with central angle 60 if
the radius of the circle is 3 m.
Solution:
To use the formula for the area of a circular sector, we
must find the central angle of the sector in radians:
60° = 60( /180) rad =  /3 rad.
Thus, the area of the sector is
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Circular Motion
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Circular Motion
Suppose a point moves along a circle as shown in Figure
12. There are two ways to describe the motion of the point:
linear speed and angular speed.
Linear speed is the rate at which the distance traveled is
changing, so linear speed is the distance traveled divided
by the time elapsed.
Figure 12
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Circular Motion
Angular speed is the rate at which the central angle  is
changing, so angular speed is the number of radians this
angle changes divided by the time elapsed.
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CIRCULAR MOTION
• CONSIDER POINT P TRAVELING THROUGH π/3
RADIANS ON A CIRCLE OF RADIUS 2 CM IN 10 SEC
A) Find the linear speed:
B) Find the angular speed
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Example 6 – Finding Linear and Angular Speed
A boy rotates a stone in a 3-ft-long sling at the rate of 15
revolutions every 10 seconds. Find the angular and linear
velocities of the stone.
Solution:
In 10 s, the angle  changes by 15  2 = 30 radians. So
the angular speed of the stone is
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Example 6 – Solution
cont’d
The distance traveled by the stone in 10 s is
s = 15  2 r = 15  2  3 = 90 ft.
So the linear speed of the stone is
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Example 7 – Finding Linear Speed from Angular Speed
A woman is riding a bicycle whose wheels are 26 inches in
diameter. If the wheels rotate at 125 revolutions per minute
(rpm), find the speed at which she is traveling, in mi/h.
Solution:
The angular speed of the wheels is
2  125 = 250 rad/min.
Since the wheels have radius 13 in. (half the diameter), the
linear speed is
v = r = 13  250  10,210.2 in./min
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Example 7 – Solution
cont’d
Since there are 12 inches per foot, 5280 feet per mile, and
60 minutes per hour, her speed in miles per hour is
 9.7 mi/h
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