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2.1 Angles: Radian & Degree Measure
Homework: 2.1: 3, 4a, 6-8, 23-33 odd, 37, 38, 43,
44, 48-50, 53, 54, 58, 71-78, 91, 92, 101, 102
Read Section 2.2
Terminology
Initial side
Terminal side
Vertex
Direction in standard position
Drawing Angles: (Examples in various Quadrants)
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2.1 Angles: Radian & Degree Measure
DEGREE MEASURE
One full revolution corresponds to 2π radians or 360º.
1  60'

1'  60' '
Example: Convert 19.256º to Degrees minutes seconds
Example: Convert 53º23’15’’ to decimal degrees
DEF: Radian
One radian is the measure of a central angle that intercepts
an arc s equal in length to the radius of a circle
Common angles:
Right
Straight
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Common
2.1 Angles: Radian & Degree Measure
IMPORTANT: Any angle measured in degrees MUST
have a degree symbol. An angle without a º in ALWAYS in
radians!!!
180º = π radians
Converting between degrees and radians
a) º  radians: multiply by  radians
180
b) radians  º : multiply by
180
 radians
Example: Convert the following to Radians
a) 120º
b) -20º
c) 240º
d) 165º
e)  º
6
f) 
6
NOTE: All answers should be given as exact fractions
(NOT DECIMALS). Do not approximate π.
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2.1 Angles: Radian & Degree Measure
Example: Convert the following to Degrees:
a) 
6
b)  7
5
c) 9
20
d) 27
On Calculator:
From Radians to Degrees
 Put Calculator in Degree Mode
 Input (angle)r
From Degrees to Radians
 Put calculator in Radian Mode
 Input (angle)°
If you divide by π  Frac will give you the fraction
coefficient.
DMS converts to Degrees, Minutes, Seconds
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2.1 Angles: Radian & Degree Measure
Sketch the following angles. (Note you must carefully
watch the mode!)
11
3
2
29
6
10

4
3
10°
Watch out for common mistakes! If the angle is negative,
you must draw it clockwise. If the magnitude angle is
greater than 2π, you must draw cycles around. If there is no
degree symbol, the angle MUST be in radians!
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2.1 Angles: Radian & Degree Measure
ACR LENGTH: s  r
Note: θ must be in radians!!! Units of s and r macth.
Example: A circle has a diameter of 10 inches. Find the
length of the arc intercepted by an angle of 120º
Example: Find the measure of the central angle of a circle
of radius r = 20 ft that intercepts an arc of length s = 100 ft
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2.1 Angles: Radian & Degree Measure
Example: You wish to cut a piece of pizza from a 12
inch pizza with the crust measuring 4.7 inches. In
order to cut your pizza accurately, you needs to find
the measure of the angle formed by the two non-crust
sides of the pizza. Find the measure of the angle in
degrees of the angle formed by the two non-crust
sides of the pizza. Give your answer accurate to 4
decimal places.
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2.1 Angles: Radian & Degree Measure
Example: As the large hand on a circular clock moves
from 12:40 pm to 1:00 pm it sweeps out a distance of
18 in. How big is diameter of the clock?
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2.1 Angles: Radian & Degree Measure
Distance between cities: The latitude of a location L
is the angle formed by a ray drawn from the center of
the Earth to the equator and a ray drawn from the
center of the Earth to L.
Example: Charleston, West Virginia is due west of
New Orleans, Louisiana. Find the distance between
Charleston (35º30’ north latitude) and New Orleans
(29º30’ north latitude). Assume the radius of the
earth is 3960 miles.
Watch out for common mistakes! The angle MUST
be in radians BEFORE plugging into the formula.
You need to convert angles to degrees AFTER using
formula.
Always include units!
Page 9 of 10
2.1 Angles: Radian & Degree Measure
Review:
Angle conversions
180º = π radians
1  60'
1'  60' '

Example: Convert 12º15’45” to radians
Example: Convert 5 to degrees, minutes seconds
7
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