4.1

An angle is formed by two rays that have a common endpoint called the vertex . One ray is called the initial side and the other the terminal side. The arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side.
C
è
A
Initial Side Terminal Side
B
Vertex

Angles of the Rectangular Coordinate System
An angle is in standard position if
• its vertex is at the origin of a rectangular coordinate system and
• its initial side lies along the positive x-axis.
y y
 is positive
Terminal
Side  Vertex
Initial Side x x
Vertex

Initial Side
Terminal
Side
 is negative
Positive angles rotate counterclockwise .
Negative angles rotate clockwise .

The figures below show angles classified by their degree measurement. An acute angle measures less than 90º. A right angle, one quarter of a complete rotation, measures 90º and can be identified by a small square at the vertex. An obtuse angle measures more than 90º but less than 180º. A straight angle has measure 180º.
180º
90º


Acute angle
0º < 
< 90º
Right angle
1/4 rotation
Obtuse angle
90º < 
< 180º
Straight angle
1/2 rotation

An angle of x
º is coterminal with angles of x
º + k ·
360º where k is an integer.

Assume the following angles are in standard position. Find a positive angle less than 360º that is coterminal with: a. a 420º angle b. a –120º angle.
Solution We obtain the coterminal angle by adding or subtracting 360º.
Our need to obtain a positive angle less than 360º determines whether we should add or subtract.
a. For a 420º angle, subtract 360º to find a positive coterminal angle.
420º – 360º = 60º
A 60º angle is coterminal with a 420º angle. These angles, shown on the next slide, have the same initial and terminal sides.

420º y
y
240º
60º x
-120º x
Solution b. For a –120º angle, add 360º to find a positive coterminal angle.
-120º + 360º = 240º
A 240º angle is coterminal with a –120º angle. These angles have the same initial and terminal sides.

• Find and graph positive and negative coterminal angle for 175 degrees.
535 degrees
-185 degrees

• For an x
º angle, the complement is a 90º – x
º angle. Thus, the complement’s measure is found by subtracting the angle’s measure from 90º.
• For an x
º angle, the supplement is a 180º – x
º angle. Thus, the supplement’s measure is found by subtracting the angle’s measure from 180º.

•
One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle.

Consider an arc of length s on a circle or radius r.
The measure of the central angle that intercepts the arc is

= s/r radians.
s
 r
O r

• Using the basic relationship  radians =
180º,
• To convert degrees to radians, multiply degrees by (
 radians) / 180

• To convert radians to degrees, multiply radians by 180

/ (
 radians)

Convert each angle in degrees to radians
40º
75º
-160º

Solution:
• 40º = 40* 
/180 = 2

/9
• 75º = 75* 
/180 = 5

/12
• -160º = -160* 
/180 = -8

/9


4

6

5
7

12
180 degrees
45 degrees

180


216 degrees
105 degrees

Let r be the radius of a circle and
 the nonnegative radian measure of a central angle of the circle. The length of the arc s intercepted by the central angle is s = r
 
O r

A circle has a radius of 7 inches. Find the length of the arc intercepted by a central angle of 2

/3
Solution: s = (7 inches)*(2

/3)
=14

/3 inches

• A clock has a minute hand that is 3 inches long. Find the distance the tip traveled in 10 minutes.

If a point is in motion on a circle of radius r through an angle of
 radians in time t , then its linear speed is v = s/t where s is the arc length given by s = r its angular speed is

=

/t

, and

• A wind machine used to generate electricity has blades that are 10 feet in length. The propeller is rotating at 4 revolutions per second. Find the linear speed, in feet per second, of the tips of the blades.
 
2
 
4
1

8
 rad sec
First we find angular speed.
The angle is 2pi and it does 4 revolutions in 1 sec.
Next we find linear speed by multiplying angular speed times radius v

8
 
10
1

80
 ft sec

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