Transcript Section 5.1 Angles and Arcs

Section 5.1
Angles 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
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
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.

Vertex
Initial Side
Counterclockwise rotation
Positive Angle

Vertex
Clockwise rotation
Initial Side
Negative Angle

Vertex
Initial Side
Counterclockwise rotation Positive Angle
An angle  is said to be in standard position
if its vertex is at the origin of a rectangular
coordinate system and its initial side
coincides with with positive x - axis.
Terminal
side
y

Vertex
Initial side
x
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
y

 is a quadrantal angle
x

 lies in Quadrant III
x
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 360 .
Terminal side
Initial side
Vertex
1
One degree, 1 , is
revolution.
360

1
A right angle is an angle of 90 , or
4
revolution.

Terminal
side
Vertex
Initial side
1
90 angle;
revolution
4

A straight angle is an angle of 180 ,
1
or revolution.
2
Terminal side
Vertex
Initial side
1
180 angle; revolution
2


Draw a -135 angle.
y
Vertex
Initial side
x
 135

One minute, denoted, 1 , is defined as
1
degree.
60
One second, denoted, 1 , is defined as
1
1
second, or
degree.
60
3600

1 counterclockwise revolution = 360
60 = 1

60 = 1
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

r
1 radian
For a circle of radius r, a central angle of
radians subtends an arc whose length s is
s  r
Find the length of the arc of a circle of radius
4 meters subtended by a central angle of 2
radians.
r  4 meters and  = 2 radians
s  r  42  8 meters
1 revolution = 2 radians
180   radians

1 degree =
1 radian =

180
180

radian
degrees
Convert 30 1255 to a decimal in degrees.

1
1 

30 1255   30  12   55 

60
3600 


 30  0.2  0.015278

 30.215278


Convert 45.413 to D MS form.


60
0.413  0.413    24.78
1
60
0.78  0.78 
 46.8  47
1


45.413  45 2447


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
s
v
t
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.
 

t
To find relation between angular
speed and linear speed, consider the
following derivation.
s  r
s  r
s r
 r
s
t  t
t
vt  r
v  r
Acknowledgement
Thanks to Addison Wesley and Prentice
Hall.
These notes are taken from
Sullivan Algebra and Trigonometry