Measuring Angles
Angle:
An angle consists of two rays (half-line) that originate at the same point (vertex).
One ray is called the initial side (often places on the x-axis).
The other ray is called the terminal side (the side that revolved from the initial side).
For out purposes, we will place the vertex of an angle on the origin of the coordinate axes
and the initial side on the positive side of the x-axis
Systems of Measurement
There are two units that can be used for measuring angles radians and degrees.
The radian measure of an angle with vertex at the center of a unit circle is the length of
the arc made by the angle. (You may need to define what a unit circle is.)
(ask the students to figure out how far it is all the way around, half way around, ¾ of the
way around, ¼ of the way around)
An angle formed by one complete rotation, by definition, consists of 360 equal parts
called degrees.
Conversion between the two systems



Ask students how you might convert between radians and degrees.
Encourage discussion of all the way around being 360 degrees and also being 2.
How can we use this to find a way to convert from one system to another?
Examples:
 Convert 75 degrees to radians.
 Convert 60 degrees to radians.
 Convert 135 degrees to radians.
 Convert 150 degrees to radians.
 Convert 40 degrees to radians.
 Convert 25 degrees to radians.
 Convert 300 degrees to radians.
 Convert 72 degrees to radians.
 Convert 270 degrees to radians.
(also ask where these are on the unit circle)

radians to degrees.
5
3
 Convert
radians to degrees.
4

Convert

Convert 

radians to degrees.
6
11
 Convert 
radians to degrees.
6
2
 Convert
radians to degrees.
3
9
 Convert
radians to degrees.
2
 Convert 7 radians to degrees.
(also ask where these are on the unit circle)
 Complete the table below
degrees 30
radians
9
0.45 120
4
-100
2
Positive and Negative Measurements
For any real number t, let P(t) be the point on the unit circle for which the arc of the
circle from (1, 0) to P(t) is |t| units, the distance is measured in the counterclockwise
direction when t  0 , and in the clockwise direction when t  0 .
For any real number t, the angle generated by rotating counterclockwise from the positive
x-axis to the point P(t) on the unit circle has a radian measure t.
Coterminal or coincides
Ask students what they think it means for two angles to be coterminal.
Two angle are coterminal (of coincide) if the terminal sides of the two angles are the
same.
Examples:
 Find the radian measure of four angles in standard position that are coterminal

with the angle  radians when in standard position
6







Find the radian measure of an angle in standard position that has measure between
19
0 and 2 that is coterminal with the angle
radians in standard position.
4
Determine if the angles at 45 degrees and 405 degrees coincide.
Determine if the angles at 120 degrees and 570 degrees coincide.

13
Determine if the angles at
radians and
coincide.
6
6
3
7
Determine if the angles at
radians and 
coincide.
4
4
Determine if the angles at 150 degrees and 240 degrees coincide
2
4
Determine if the angles at
radians and 
coincide.
3
3
Points on a unit circle
We are concerned with the connection between a real number t and the unique arc of the
unit circle described by the point P(t). The coordinates of the point P(t) are (x(t), y(t)).
Examples:


Show the approximate location on the unit circle of P(t) for t 


Show the approximate location on the unit circle of P(t) for t   .
Show the approximate location on the unit circle of P(t) for t  2 .
3
Show the approximate location on the unit circle of P(t) for t 
.
2
5
Show the approximate location on the unit circle of P(t) for t 
.
6
3
Show the approximate location on the unit circle of P(t) for t 
.
4
4
Show the approximate location on the unit circle of P(t) for t 
.
3
7
Show the approximate location on the unit circle of P(t) for t 
.
6

Show the approximate location on the unit circle of P(t) for t   .
4
4
Show the approximate location on the unit circle of P(t) for t  
.
3
37
Show the approximate location on the unit circle of P(t) for t  
.
6
7
Show the approximate location on the unit circle of P(t) for t  
.
4









2
.

















21
.
2
317
Show the approximate location on the unit circle of P(t) for t 
.
4
33
Show the approximate location on the unit circle of P(t) for t  
.
2
19
Show the approximate location on the unit circle of P(t) for t  
.
6
3 4
If P(t) has coordinates  ,  , find the coordinates of P(t   ) .
5 5
3 4
If P(t) has coordinates  ,  , find the coordinates of P(t ) .
5 5
3 4
If P(t) has coordinates  ,  , find the coordinates of P(t   ) .
5 5
3 4
If P(t) has coordinates  ,  , find the coordinates of P (t   ) .
5 5
3
7
 , find the coordinates of P(t  2 ) .
If P(t) has coordinates  ,

4
4


3
7
 , find the coordinates of P(t ) .
If P(t) has coordinates  ,

4
4


3
7
 , find the coordinates of P(t   ) .
If P(t) has coordinates  ,

4
4


3
7
 , find the coordinates of P(t  3 ) .
If P(t) has coordinates  ,

4
4



5 2
,  , find the coordinates of P(t   ) .
If P(t) has coordinates  
3
3


5 2
,  , find the coordinates of P(t ) .
If P(t) has coordinates  
3
3


5 2
,  , find the coordinates of P(t   ) .
If P(t) has coordinates  
3
3


5 2
,  , find the coordinates of P ( t   ) .
If P(t) has coordinates  
3
3


13
13 
 , find the coordinates of P(t  2 ) .
, 3
If P(t) has coordinates   2

13
13


Show the approximate location on the unit circle of P(t) for t 




If P(t) has coordinates   2


If P(t) has coordinates   2


If P(t) has coordinates   2

13
13 
 , find the coordinates of P(t ) .
, 3
13
13 
13
13 
 , find the coordinates of P(t   ) .
, 3
13
13 
13
13 
 , find the coordinates of P ( t   ) .
, 3
13
13 
Length of a Circular Arc
Ask the students how they would find the length of an arc for a circle of radius r and
angle measure .
In a circle of radius r, the length s of an arc with angle  radians is s  r .
Examples:
 Find the length of an arc with angle 45 degrees in a circle with radius 8 meters.
 Find the length of an arc with angle 30 degrees in a circle with radius 2 miles.
 In a circle of radius 3 inches and arc of length 6 inches has angle . Find the
measure of the angle in degrees and radians.
 In a circle of radius 5 miles and arc of length 3 miles has angle . Find the
measure of the angle in degrees and radians.
 Find the length of the arc in the figure
s
4
110

Find the radian measure of the angle in the figure
9

4
Area of a Circular Sector:
Ask students how to find the area of a circular sector formed by an angle of  radians in a
circle of radius r.
In a circle of radius r, the area A of a circular sector formed by an angle of  radians is
1
A  r 2 .
2
Examples





Find the area of the circular sector with angle
meters.
Find the area of the circular sector with angle 30 degrees in a circle of radius 4
inches.
The area of a sector of a circle with an angle 45 degrees is 2 square feet. Find the
radius of the circle.
The area of a sector of a circle with radius 3 miles is 10 square miles. Find the
angle of the sector.
Find the area of the sector in the figure
4
65

1
radians in a circle of radius 10
2
Find the area of the sector in the figure
1.5 rad
6
Angle of rotation
Angle of rotation = (number of revolutions) * (2)
Angular Speed
Angular speed = (number of revolutions per time) * (2)
Linear Speed
Linear speed = (radius) * (angular speed)
General Examples:





Determine the positive radian measure of the angle that the second had of a clock
traces out in 2 minutes and 15 seconds.
A wheel in a large machine is 2.8 feet in diameter and rotates at 1200 rpm. What
is the angular speed of the wheel? How fast is a point on the circumference of the
wheel traveling in feet per minutes? How fast is a point on the circumference of
the wheel traveling in miles per hour?
Determine the distance between Syracuse, New York and Virginia Beach,
Virginia which lie on approximately the same meridian. Syracuse has latitude
approximately 43 degrees North and Virginia Beach has latitude approximately
36.5 degrees North. The radius of the earth is approximately 3960 miles. (find
the length of the arc between the two cities)
Determine the distance between New York City , United State and Bogota,
Columbia which lie on approximately the same meridian. New York City has
latitude approximately 40.5 degrees North and Bogota has latitude approximately
4.6 degrees North. The radius of the earth is approximately 3960 miles. (find the
length of the arc between the two cities)
Determine the distance between the North Pole and Springfield, Illinois which lie
on approximately the same meridian. The North Pole has latitude approximately
90 degrees North and Springfield has latitude approximately 40 degrees North.
The radius of the earth is approximately 3960 miles. (find the length of the arc
between the two cities)

A weather satellite orbits the earth in a circular orbit 500 miles above the earth’s
surface. What is the radian measure of the angle (measured from the center of the
earth) through which the satellite moves in traveling 600 miles along its circular
orbit? (Assume the radius of the earth is 3960 miles.)