Download 4.1 Radian and Degree Measure

4.1 Radian and Degree Measure
An angle AOB (notation: AOB ) consists of two rays R1 and R2 with a common vertex O (see Figures
below). We often interpret an angle as a rotation of the ray R1 onto R2 . In this case, R1 is called the
initial side of the angle, and R2 is called the terminal side of the angle. If the rotation is
counterclockwise, the angle is considered positive, and if the rotation is clockwise, the angle is
considered negative.
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
1
of a complete revolution. In
360
calculus and other branches of mathematics, a more natural method of measuring angles is used--radian (abbreviated rad) measure.
Definition of Radian: one radian is the measure of a central angle  that intercepts an arc s equal in
length to the radius r of the circle. Algebraically, this means that

where  is measured in radians.
s
r
The circumference of the circle of radius r is 2 r , it follows that a central angle of one full revolution
(counterclockwise) corresponds to an arc length of s  2 r .
Because the radian measure of an angle of one full revolution is 2 , you can obtain the following.
1
2
1
2 
1
2 
revolution=
  radians, revolution=
 radians, revolution=
 radians
2
2
4
4
2
6
6
3
These and other common angles are shown in the following.
Recall that the four quadrants in a coordinate system are numbered I,II, III, and IV. The following graph
shows which angles between 0 and 2 lie in each of the four quadrants. Note that angels between
0 and

2
are called acute angles and angles between

2
and  are obtuse angles.
Two angles are coterminal if they have the same initial and terminal sides. A given angle  has infinitely
many coterminal angles:   2n , where n is an integer.
Two positive angles  and  are complementary (complements of each other) if    

2
.
Two positive angles  and  are supplementary (supplements of each other) if      .
Examples:
a.
b.
 19
3

3
,
3
and
are coterminal? (Yes)

are complementary ? (yes)
6

4
c.
are supplementary ? (yes)
and
5
5
5
d. If possible, find the complement of
. (no complement)
8
The second way to measure angles is in terms of degrees, denoted by the symbol  . A measure of one
degree ( 1 ) is equivalent to a rotation of
1
of a complete revolution about the vertex. To measure
360
angles, it is convenient to mark degrees on the circumference of a circle. A full revolution
(counterclockwise) corresponds to 360 , a half revolution to 180 .
Conversions between degrees and radians.
a. To convert degrees to radians, multiply degrees by
 rad
180
180
b. To convert radians to degrees, multiply radians by
 rad
Examples: Converting degrees to radians or radians to degrees.
a.
 
30  ?  
6
b.
 
45  ?  
4
c.

2
 ?  90
d.   ? 180
e.
f.
5
 ? 135
4
 3 
270  ?  
 2 
The radian measure formula,  
s
, can be used to measure arc length s given by
r
s  r
Arc Length. For a circle of radius r , a central angle  intercepts an arc of length s given by
s  r
where  is measured in radians.
Example: A circle has a radius of 2 inches. Find the length of the arc intercepted by a central angle of
240 . (answer:
8
)
3
Linear and Angular Speeds. Consider a particle moving at a constant speed along a circular arc of radius
r . If s is the length of the arc traveled in time t , then the linear speed v of the particle is
Linear speed v 
arc length s

time
t
Moreover, if  is the angle (in radian measure) corresponding to the arc length s , then the angular
speed  of the particle is
Angular speed  
central angle 

time
t
Example: The second hand of a clock is 10.2 centimeters long, as shown in the following graph. Find the
linear speed of the tip of this second hand as it passes around the clock face.
Solution: Linear speed 
s
2 r
2 (10.2 centimeters)


 1.068 centimeters per second
t 60 seconds
60 seconds
Example. The blades of a wind turbine are 116 feet long. The propeller rotates at 15 revolutions per
minute.
a. Find the angular speed of the propeller in radians per minute.
b. Find the linear speed of the tips of the blades.
Solution: Angular speed 
Linear speed 

t

30 radians
 30 radians per minute
1 minute
s r 116  30 feet


 10,933 feet per minute
t
t
1 minute
A sector of a circle is the region bounded by two radii of the circle and their intercepted arc.
Area of a Sector of a circle. For a circle of radius r , the area A of a sector of the circle with central
angle  is given by
1
A   r2
2
Example: A sprinkler on a golf course fairway sprays water over a distance of 70 feet and rotates
through an angle of 120 . Find the area of the fairway watered by the sprinkler.
Solution:   120 
2
radians
3
1
1 2
A   r2  
 702  5131 square feet
2
2 3