Section 5.1 Angles and Arcs

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Section 5.1Angles and Arcs

• Objectives of this Section
• Convert Between Degrees, Minutes, Seconds, and
  Decimal Forms for Angles
• Find the Arc Length of a Circle
• Convert From Degrees to Radians, Radians to
  Degrees
• Find the Linear Speed of Objects in Circular
  Motion

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A ray, or half-line, is that portion of a line
that starts at a point V on the line and extends
indefinitely in one direction. The starting
point V of a ray is called its vertex.
V
Ray
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If two lines are drawn with a common vertex, they
form an angle. One of the rays of an angle is
called the initial side and the other the
terminal side.
Terminal side
Initial Side
Vertex
Counterclockwise rotation
Positive Angle
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Terminal side
Vertex
Initial Side
Clockwise rotation Negative Angle
Terminal side
Vertex
Initial Side
Counterclockwise rotation Positive Angle
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y
Terminal side
x
Initial side
Vertex
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When an angle is in standard position, the
terminal side either will lie in a quadrant, in
which case we say lies in that quadrant, or
it will lie on the x-axis or the y-axis, in which
case we say is a quadrantal angle.
y
x
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Angles are commonly measured in either Degrees or
Radians
The angle formed by rotating the initial side
exactly once in the counterclockwise direction
until it coincides with itself (1 revolution) is
said to measure 360 degrees, abbreviated
Terminal side
Initial side
Vertex
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Terminal side
Vertex
Initial side
9
Initial side
Terminal side
Vertex
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y
Initial side
Vertex
x
Terminal side
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Consider a circle of radius r. Construct an
angle whose vertex is at the center of this
circle, called the central angle, and whose rays
subtend an arc on the circle whose length is r.
The measure of such an angle is 1 radian.
r
1 radian
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For a circle of radius r, a central angle of
radians subtends an arc whose length s is
Find the length of the arc of a circle of radius
4 meters subtended by a central angle of 2
radians.
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Suppose an object moves along a circle of radius
r at a constant speed. If s is the distance
traveled in time t along this circle, then the
linear speed v of the object is defined as
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Let (measured in radians) be the the central
angle swept out in time t. Then the angular
speed of this object is the angle (measured
in radians) swept out divided by the elapsed time.
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To find relation between angular speed and
linear speed, consider the following derivation.
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Acknowledgement

• Thanks to Addison Wesley and Prentice Hall.
• These notes are taken from
• Sullivan Algebra and Trigonometry