Lesson 5-1 Angles and Degree
Measure
Objective: To convert decimal
degrees to measure degrees.
To find the number of degrees in a
given number of rotations.
Identify coterminal angles.
Trigonometry
The branch of mathematics that
studies triangles. It deals with the
relationship between the sides and
the angles.
In order to do this you must also
understand the relationship between
angles and circles.
Angles
• An angle is formed by two rays with
a common endpoint. (geometry
definition)
• An angle is the result of a rotation
of a ray about its endpoint.
(trigonometry definition)
Standard Position
When the initial side is on the positive x
axis and the endpoint is on the origin
Then the angle is in standard position.
Positive Angle – Standard
Form
α
Negative Angle – Standard
Form
β
Quadrant I
• Where the terminal side lays is
where the angle is said to lie.
Between 0o and
90o or 0 and 
2
Quadrant II
Between 90o and
180o or  and
2

Quadrant III
Between 180o and
3
o
270 or and 2

Quadrant IV
Between 270o and
3
o
360 or 2 and 2
Quadrantal
• When the terminal side lies on an
axis it is called a quadrantal.
Angular Measurement
1
360
• Degree of a complete rotation
in the counterclockwise direction.
1o
Decimal-Degrees and
Degree-Minute-Second Form
• There are 2 forms for expressing
degrees:
– decimal degrees
– degree-minute-second
Basic Conversions:
• 1 degree= 60
minutes
• 1o = 60’
• 1 minute = 60
seconds
• 1’=60”
• 1 degree=3600
seconds
• 1o = 3600’’
Example
Convert 0.5o to Minutes
• 0.5o
0.5 Degrees
 (0.5  60' )
1 degree=60 minutes
 30'
30 minutes
Example:
Convert 38.427o to Degree-MinuteSecond
38.427
38.427 degrees
38  .427
38  (.427  60)' 1 degree=60 min.
38 degrees 25.62
38  25.62'
minutes
38  25'.62'
38  25'(.62  60)"1 min. = 60 sec.
38  25'37.2"
38 25' 37.2"
Example:
Convert 30’ to Decimal
Degrees
30'
30'

60
 .5
• 1 minute = 1/60
degree
Example:
Convert 28o 32’ 45” to Decimal
Degrees
28 32' 45"
28 degrees, 32
minutes, 45 seconds
 32   45 
28   '  
"
 60   3600 
 28.546
1 minute=1/60 degree
1 second=1/3600
degree
Coterminal Angles
• Coterminal – two angles that have a
terminal side that is in the same
position. (ex: +30o and -330oor 60o and
420o)
Coterminal Angles
• Angle θ is coterminal with every angle:
• θ + 360on
• n is an integer that tells how many
times around the circle.
Examples
• Find the positive angle that is
coterminal with each of the
following:
• 430o
• 25o
70o
385o
Reference Angle
• The reference angle θʹ (theta prime) for
the angle θ is the acute angle that is
formed by the terminal side of the angle
and the x axis. (an acute angle means
that it must be between 0o and 90o.
θ
θ
θʹ
θʹ
Reference Angle
• If θ is an acute angle and lies in Quadrant I then
the reference angle is the same as θ. (θʹ = θ)
• If θ lies in Quadrant II then θ + θʹ = 180o.
• If θ lies in Quadrant III then θ – θʹ = 180o.
• If θ lies in Quadrant IV then θ + θ ʹ = 360o.
Reference Angles
• Find the reference angle if θ is:
160o
30o
-95o
125o