Angles and Radian Measure
4.1 – Angles and Radian Measure
An angle is formed by rotating a ray around its
endpoint. The original position of the ray is called the
initial side of the angle, and the ending position of the
ray is called the terminal side. The endpoint of the ray
is called the vertex of the angle.
vertex
4.1 – Angles and Radian Measure
An angle is said to be in standard position if its vertex is
at the origin, and its initial side is on the positive x-axis.
An angle is positive if it is rotated counterclockwise,
negative if it is rotated clockwise.
positive
negative
4.1 – Angles and Radian Measure
Degree Measure
90º (right angle)
360º
180º
(straight angle)
270º
4.1 – Angles and Radian Measure
Degree Measure
90º
obtuse
(between 90º and 180º)
180º
acute
(between 0º and 90º)
360º
270º
Converting from Degrees-Minutes-Seconds
(DMS) to Decimal Degrees (DD)
 “DMS” stands for Degrees-Minutes-Seconds.
 The conversions are based on subdivisions
of a whole degree where…
 1 degree = 60 minutes
 1 minute = 60 seconds
It follows that 1 degree = 3600 seconds
From DMS to DD
M
S
D M ' S "  D 

60 3600
From DD to DMS
1. Multiply the decimal part of the degrees
(to the right of the decimal point) by 60.
These are your minutes.
2. Now multiply the decimal part of your
minutes (to the right of the decimal point)
by 60. These are your seconds.
*There is a DMS button on your calculator.*
Examples
 35 degrees 23 minutes 57 seconds to DD
3523'57"
(round to four decimal places)
 97.554 degrees to DMS
4.1 – Angles and Radian Measure
If θ is an angle with its vertex at the center of a circle of
radius r, and θ intercepts an arc of length s on that circle,
then θ = s radians.
r
s

θ
r
4.1 – Angles and Radian Measure
If s is the length of an arc intercepted by a central
angle θ (in radians) in a circle of radius r, then s = rθ.
s
θ
r
4.1 – Angles and Radian Measure
If θ intercepts an arc of length equal to the radius of the
circle, θ has a measure of 1 radian.
r
θ
r
4.1 – Angles and Radian Measure
One complete rotation
s
θ
r
s = circumference = 2πr
2πr
θ = r = 2π radians
4.1 – Angles and Radian Measure
Radian Measure
π (right angle)
2
180 = π
(straight angle)
2π
3π
2
4.1 – Angles and Radian Measure
π
To convert from degrees to radians, multiply by
.
180
180
To convert from radians to degrees, multiply by
.
π
 π radians  π
  radians
60º = 60 degrees 
 180 degrees  3
3
30
5π
 180 degrees 
radians 
 = 150º
6
 π radians 
 180 degrees 

 π radians 
5 radians 
= 286.48º
Examples
 Convert the angle in degrees to radians.
a ) 18
b )  250
 Convert the angle in radians to degrees.
11
a)
6
5
b) 
2
4.1 – Angles and Radian Measure
Two angles are said to be coterminal if they have the
same terminal side.
40º
Angle = 40º
Coterminal angle: 40º − 360º = −320º
40º + 360º = 400º
40º + 720º = 760º
4.1 – Angles and Radian Measure
Two angles are said to be coterminal if they have the
same terminal side.
π
6
Angle = π 6
Coterminal angle: π 6  2π  11π 6
π
π
6
6
 2 π  13π
 4 π  25π
6
6
4.1 – Angles and Radian Measure
A clock has a minute hand which is 8 inches long. If
the minute hand moves from 12:00 to 3:00, how far
does the tip of the minute hand move?
s = rθ = (8)(90) = 720 inches
 2  = 4π = 12.57 inches
s = rθ  (8) π
4.1 – Angles and Radian Measure
Two angles whose sum is 90º are called complementary.
Two angles whose sum is 180º are called supplementary.
40º
Angle = 40º
Complement: 90º − 40º = 50º
Supplement: 180º − 40º = 140º