Angles and Radian Measure
Section 4.1
Objectives
• Estimate the radian measure of an angle
shown in a picture.
• Find a point on the unit circle given one
coordinate and the quadrant in which the
point lies.
• Determine the coordinates of a point on the
unit circle given a point on the unit circle.
• Find coterminal angles.
• Convert angle measures between radians and
degrees.
• Determine the linear speed of an object
traveling in a circular motion.
• Determine the arc length on a given circle.
Vocabulary
•
•
•
•
•
•
•
•
•
unit circle
radian measure of an angle (radians)
degrees
vertex of an angle
terminal side of an angle
initial side of an angle
linear speed (length per unit of time)
length of a circular arc
angular speed (radians per unit of time)
Unit Circle
Consider the picture below. The
angle θ is an integer when
measured in radians. Give the
measure of the angle.
The angle that is
straight up (right) is
approximately 1.571
radian. The straight
angle is approximately
3.142 radians. Since
this angle is closer to
the straight angle than
to the right angle, the
radian measure would be
about 3 radians.
Coterminal Angles:
Angles are coterminal if they are
in standard position and have the
same terminal side.
Find an angle between 0
and 2π that is coterminal
to the angle
77
3
Since this angle is positive, we need to subtract multiples of 2π to find a
coterminal angle. To determine how many multiples, we can start by
dividing 77 by 3. This will tell us how many half circles there are. Since
F
we need full times around
the circle, we need to divide that number by 2
for how many multiples
i of 2π we need to subtract.
77
3
n
d
continued on next slide
Coterminal Angles:
Angles are coterminal if they are
in standard position and have the
same terminal side.
Find an angle between 0
and 2π that is coterminal
to the angle
77
3
77  3  25.66667 25 full half-ways around a circle
25  2  12.5F full times circles
77
3
i
77
77 362  77 72 5
 122   n 



3
3
3
3
3
3
d
Formulas
• Conversion between degrees and radian
• Length of a circular arc
s  r
where the angle measure is
in non  negative radians
• Linear speed
s
v 
t
where s  r
• Angular speed   
t
Find the degree measure of an
2
angle with radian measure 3
To convert from radians to degrees, we want to
multiply the radian measure by a fraction made
up of radians and degrees equal to 1. Since we
are trying to get rid of the radians, we will need
the fraction to have radians in the denominator.
Since 180 degrees is the same as π radians, we
will use these two numbers in our fraction.
2 radians 180 degrees

 120 degrees
3
 radians
Find the radian measure of an
angle with degree measure  100
To convert from degrees to radian, we want to
multiply the degree measure by a fraction made
up of radians and degrees equal to 1. Since we
are trying to get rid of the degrees, we will
need the fraction to have degrees in the
denominator. Since 180 degrees is the same as
π radians, we will use these two numbers in our
fraction.
100 degrees
 radians
100


radians
1
180 degrees
180
Find the length of the arc on a circle
of radius r = 6 inches intercepted by
a central angle θ = 135 degrees
For this problem, we will use the formula for the length
of a circular arc.
s  r
where the angle measure is
in non  negative radians
In order to do this, we must change the angle
measure to radians.
135 degrees
 radians
135


radians
1
180 degrees
180
Now we can plug this into the formula to get
s  6*
135
inches  14.13716694 inches
180
A Ferris wheel has a radius of 30
feet and is rotating at 3.5
revolutions per minute. Find the
linear speed, in feet per minute,
of a seat on the Ferris wheel.
For this we will need the linear speed formula.
v 
s
t
where s  r
We will need to calculate s (the length of the circular arc that the
Ferris wheel goes through and find the time t that is takes to go
through that arc.
The Ferris wheel goes through 3.5 revolutions in one minute. This
means that the angle is 3.5 times around the circle. Since one time
around the circle is 2π radians, we need to multiply 3.5 by 2π to
find the radian measure of the angle.
continued on next slide
A Ferris wheel has a radius of 30
feet and is rotating at 3.5
revolutions per minute. Find the
linear speed, in feet per minute,
of
a
seat
on
the
Ferris
wheel.
This will give us a radian angle measure of 7π. We now use
that in the formula for s to find the length of the circular
arc.
s  30 * 7  210 feet
Now the amount of time that the Ferris wheel took to go that
210π feet was 1 minute. This means that to find the linear speed,
we divide the distance by the time it took to travel that distance.
This will give us:
v 
210 feet
 659.7345 feet per minute
1 minute