4.1 Notes
Radians and Degree Measure
Definitions:
TRIGONOMETRY:
An angle is formed by rotating a half-line called a ray around its endpoint.
The initial side of the angle remains fixed. A second ray called the terminal side of the
angle starts in the initial side position and rotates around a common endpoint called the
vertex until it reaches its terminal position.
Terminal side
Vertex
Initial side
When a terminal side is rotated counterclockwise, the angle is positive.
When a terminal side is rotated clockwise, the angle is negative.
Two different angles having the same initial AND terminal sides are said to be
coterminal.
Sketching angles in standard position:
To position an angle in standard position, start at 0°. (EAST)
To draw a positive angle, move counterclockwise. (UP)
To draw a negative angle, move clockwise. (DOWN)
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4.1 Notes
Radians and Degree Measure
EX 1:
 Sketch the following in standard position:
 Then find 1 positive and 1 negative coterminal angle.
 Then, state in which quadrant the angle lies
a)   125
c)   212
b)   315
d)   146
Types of Angles:
Acute:
Standard Units of Measure:
Degree
Right:
Radian
Obtuse:
Complete revolution = 360°
1
1° =
of a complete revolution
360
Straight:
Complementary:
Supplementary:
EX2: Find the complement and supplement of  
2

5
4.1 Notes
Radians and Degree Measure
Unit Conversion:
Decimal Degrees: 123.68
Degree Minute Second Degrees: 2453'02"

Decimal to Degree-Minute-Second

Degree-Minute-Second to Decimal
Type in # of degrees, then
to get degree symbol
Type in # of minutes, then
to get minute symbol
Type in # of seconds, then
EX2:
a) 48.325
to get second symbol
b) 18324'55"
DAY 2
Radian-Degree Conversion
Radian: A central angle with arc length equal to 1 radius is called 1 radian
(GSP:Trig/Unit Circle)
Since C  2R
360  2 (1) radians OR 2 radians, and
180   radians
2  6.28 There are just over 6 radius lengths in a circumference of a circle
Use the circles below to label the angles in degrees and in radian measure.
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4.1 Notes
Radians and Degree Measure
Converting between degrees and radians:
Multiply by a fraction that is equal to 1.
180 
(getting rid of radians)
 ( rads )
 (rads )
To convert from degrees to radians: multiply by
(getting rid of ° notation)
180
To convert from radians to degrees: multiply by
EX 3: Convert from degrees to radians or radians to degrees
1. 18° =
2. 1.6 radians =
3.
Sketching angles (radians) in standard position:
EX 4:
 Sketch the following in standard position:
 Then find 1 positive and 1 negative coterminal angle.
 Then state the quadrant in which the lies
5

a)
b) 
6
3
c) 5 rad
d) -4 rad
4
5

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4.1 Notes
Radians and Degree Measure
Day 3
Angles and Arcs:
Given an arc RQ of a circle with center P, the angle RPQ is said to be the central angle
that is subtended by the arc RQ and vice versa.
Q
Proportions relating central angles and arcs:

360


s
C
360
s = arc length
R = radius

360

s
C
360
C = circumference
A = area of a sector
EX 5:
a) C  72" , s  12"
find 


s
2R

A
R 2
 = angle in degrees
b)   15 , s  2.5"
find r
Solving for  gives you:
  360 
s
C
Rewrite C = 2 R
  360 
s
2R
  
Use conversion to convert to radians  

 180  

360  s 

2R 180

s
R
OR you can solve for any of the variables: s  R OR
s (units)
 no(units) for 
R(units)
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4.1 Notes
Radians and Degree Measure
1
A  R 2
2
Formula for AREA of a sector
EX 6:
a) In a circle of radius 5.0 cm, find the central angle subtended by an arc length of 7 cm.
b) In a circle of radius 12 feet, find the length of an arc subtended by a central angle of
2.7 radians.
c) In a circle of radius 15 cm, find the area of the circular sector with central angle of
2
radians.
3
d) Find the distance between Atlanta( 3345' N ) and Cincinnati( 398' N ) assuming that
they fall on the same longitudinal line. The radius of the earth is 3964 mi.
Day 4
S  dis tan ce 


T  time 
The distance traveled by a point through an arc along a circular path per unit time.
m ft miles
Units you will see:
, etc.
,
,
sec sec hour
Linear Velocity:
V

 # radians 


T  time 
The change in the angle when a point travels along a circular path per unit time.
radians rads
Units you will see:
,
, etc.
sec
min
Angular Velocity:  
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4.1 Notes
Radians and Degree Measure
Relationship between linear and angular velocity:
V  R
  revs / time  2 (answer will be in rads/time)
1 revolution = 2
Diameter D = 2R
s
or V  R
AND
t
Unit Conversion with Revolutions:
 1 Revolution = 2R radians
So
V 


t
or  
V
R
EX 7:Convert 2.7 revs/min to rads/sec
EX 8:
a) An wind mill has propeller blades that are 5.0 m long. If the blades are rotating at 8
rad/sec, what is the linear velocity (to the nearest m/s) of a point on the tip of one of the
blades?
b) A point on the rim of a 6.0 ft diameter wheel is traveling at 75 ft/sec. What is the
angular velocity of the wheel in radians per second?
c) If a 6 cm shaft is rotating at 4000 rpm, what is the speed of a particle on its surface in
cm/min?
7