5.4 Radians, Arc Length, Angular
Speed
Thurs Oct 30
Do Now
Find the following:
1) Cos 150 deg
2) Tan 210 deg
HW Review: p.472 #39-69 83-105
Unit Circle
• A circle centered at the origin with a radius of
length 1 is called a unit circle
• This is useful because in a unit circle, any
reference triangles have hypotenuses of 1
Unit Circle and Arc Length
• In the unit circle, the radius is 1
• Therefore, the circumference of a unit circle is
2pi
Ex
• How far will a point travel if it goes
• A) ¼ of the way around the unit circle?
• B) 1/12 of the way?
• C) 3/8 of the way?
Ex2
• On the unit circle, mark the point determined
by each of the following real numbers
• Hint: Consider how much more or less than
1pi or 2pi we want to find
9
p
• 1)
4
• 2)
7p
6
Radian Measure
• The radian is the other measure of an angle.
• The word radian comes from the word radius
• Measuring 1 radius along the circumference of
the circle determines an angle whose measure
is 1 radian
Radians and 360 degrees
• Since 360 degrees denotes a full
circumference, and 2pi is the circumference of
the unit circle,
• 2pi radians = 360 degrees
OR
• 1pi radians = 180 degrees
Converting between degrees and
radians
• To convert from degree to radians,
deg ×
p rad
180°
= radians
• To convert from radians to degrees,
180°
radians ×
= degrees
p rad
Ex
• Convert to radians
• 1) 120 deg
• 2) -297.25 deg
Ex
• Convert to degrees
• 1) 3p
4
• 2) 8.5
Note
• When a rotation or angle is given in radians,
the word radians is optional and most often
omitted.
• If no unit is given, it is understood to be in
radians
• No degree symbol = radians!
Unit circle and radians
• Draw degrees and radians in the unit circle
here
Ex
• Find a positive angle and negative angle that
are coterminal with 2pi/3
Ex
• Find the complement and supplement of pi/6
Closure
• Convert 315 deg to radians
• Convert pi/3 to degrees
• HW: p.486 #1-59 odds
5.4 Arc Length and Central Angles,
Angular Speed
Fri Oct 31
• Do Now
• Find the following values
• 1) sin 3pi/4
• 2) cos 7pi/6
HW Review: p.486 #1-59
Radian Measure of other circles
• Radian measure can be determined using a
circle other than a unit circle
• The radian measure of a rotation of the
distance S traveled by a point at a radius R:
S
q=
R
Ex
• Find the measure of a rotation in radians
when a point 2m from the center of rotation
travels 4m
Ex
• Find the length of an arc of a circle of radius
5cm associated with an angle of pi/3
Linear and Angular Speed
• Linear speed is defined as distance traveled
per unit of time
s
v=
t
• Angular Speed is defined as amount of
rotation per unit of time
q
w=
t
Linear Speed in terms of angular speed
• The linear speed v of a point a distance r from
the center of rotation is given by
v = rw
• Where w is the angular speed in radians per
unit of time
Note
• With these problems, all units must be the
same (distance, radians, time)
• You may have to convert your given values in
order to use the previous formula
Ex
• An earth satellite in circular orbit 1200km high
makes one complete revolution every 90 min.
What is its linear speed? Let the radius of the
earth = 6400km
Ex2
• An anchor is hoisted at a rate of 2ft/sec as the
chain is wound around a capstan with a 1.8yd
diameter. What is the angular speed of the
capstan?
Ex3
• A car is traveling at a speed of 70 mph. Its tires
have an outside diameter of 28.56in. Find the
angle through which a tire turns in 10 sec
Closure
• In a circle with a 5m radius, how long is an arc
associated with an angle of 2.1 radians?
• HW: p.488 #61-79 odds
• 5.3-5.4 quiz Tues
5.3-5.4 Review
Nov 03
• Do Now
• A wheel with a 30cm radius is rotating at a
rate of 3radians/sec. What is the linear speed
of a point on its rim, in meters per minute?
69
• Through how many radians does the minute
hand of aclock rotate from 12:40 – 1:30
71
• A flywheel with a 15-cm diameter is rotating
at a rate of 7radians/sec. What is the linear
speed of a point on its rim, in cm/min
73
• A cylinder on this pres has a 21 in diameter.
The linear speed of a point on the cylinder’s
surface is 18.33 ft/sec. What is the angular
speed of the cylinder in revolutions per hour?
HW Review: p.488 #61-79
5.3-5.4 Review
• 5.3 Trig Values of any angle
– 4 Quadrants
– Reference Angles
– Unit Circle
• 5.4 Radians
– Convert from/to degrees
– Coterminal angles
– Linear and Angular speed (1)
Do Now
• What is a radian? How are radians and
degrees related?
• Length of an arc equivalent to the radius
• 2pi = 360 degrees
• Pi = 180 degrees