Transcript 4 1 Radians Degrees 03

Objectives:
1. To find linear and
angular speed
2. To find the area of a
sector
3. To convert between
DMS and decimal
degrees and use
DMS in angle
calculations
Assignment:
• P. 292: 91-94 S
• P. 292-3: 101, 103, 105,
106, 108, 113, 116
• P. 291: 71-74 S
– Also find a complement
and supplement if
possible
– Also convert to radians
• P. 291: 75-78 S
You will be able to
find linear and
angular speed
For this activity
everyone needs to
stand up and hold
your arms out
perpendicular to your
body. Now rotate
around 360° at a rate
of 45° every second.
Censored
Now that you are good
and dizzy, answer this
question: As you
were spinning
around, what had a
greater speed, your
elbows or your
fingertips? Or were
they traveling at the
same speed?
Censored
Since your fingertips and
your elbows arrived at
the same location at
the same time, and
since your fingertips
had further to travel, it
makes sense that they
traveled faster than
your elbows. This is
linear speed.
Censored
On the other hand, the
rate at which you were
rotating, π/4 radians
per second, is angular
speed.
Linear speed is how fast
your fingertips move.
Angular speed is how fast
the angle swept by
your arm changes.
Censored
The linear speed v of a particle traveling at a
constant rate along a circular arc of radius r
and length s is:
arc length s
v

time
t
Find the linear speed of
the tip of each hand
of the clock. (In case
you were wondering,
the time is 3:10:50,
and school’s nearly
out.)
The angular speed ω (omega) of a particle
traveling at a constant rate along a circular arc
of measure θ (in radians) is
central angle 


time
t
Find the angular speed
of the tip of each
hand of the clock. (In
case you were
wondering, the time
is still 3:10:50.)
Try not to vomit: A Ferris wheel with a 50-foot
radius makes 1.5 revolutions per minute. Find
the angular speed of the Ferris wheel in
rad/min and the linear speed in ft/min.
You
will be
able
to find
the
area
of a
sector
Whereas an arc was a
fraction of a circle’s
circumference, a
sector is a fraction of
a circle’s area.
A sector is like a piece
of pizza, while an arc
is just the crust.
When your central
angle is in degrees,
the area A of a sector
is
mC
A
 r 2
360
Find the area of the sector swept out a minute
hand of a clock with a radius of 9 inches over
the course of 12 minutes.
The formula for the area of a sector as learned
in geometry (and 2 slides ago) assumed the
central angle was in degrees. Convert this
formula to radians.
When the central
angle θ of a circle of
radius r is measured
in radians, then the
area A of the sector
is
1
A   r 2
2
A large pizza from
Papa John’s has a
diameter of 14 inches.
What’s the area of the
sector formed by 3
pieces of pizza if their
tips trace out an angle
measuring 2π/3
radians?
You will be able to convert between
DMS and decimal degrees and use
DMS in angle calculations
Sometimes a degree is just too
big. In that case, we can
break a degree into 60 equal
parts, called minutes (‘). For
those of you who think a
minute is too big, we can
break each of those into 60
equal parts, called seconds
(“). We’ll call angle
measures in degrees,
minutes, and seconds DMS.
Based on the previous
divisions:
1 degree = 60 minutes
1 minute = 60 seconds
33°12’14” reads 33 degrees,
12 minutes, and 14
seconds
Based on the previous
divisions:
1° = 60’
1’ = 60”
33°12’14” reads 33 degrees,
12 minutes, and 14
seconds
If there are 60 minutes in a degree and 60
seconds in a minute, how many seconds are
there in a degree?
1 degree = 3600 seconds
1° = 3600”
Although accurate, DMS are not always
convenient for calculations. In this case, we’d
want to convert DMS to decimal degrees. To
do this, just write your DMS measure as the
sum of fractions.
 12   14 
3312'14"  33      

60
3600
  

 33.2038
Convert from DMS to decimal degrees.
1. 97°4’35”
2. 13°32’50”
On the other hand, sometimes we would want to
convert decimal degrees into DMS.
32.159  329'32.4"
D
M
S
Take off the whole number. This is the
number of degrees.
32°
Multiply the remaining decimal by 60. Take (.159)(60) = 9.54
off the whole number. This is the number
9’
of minutes
Multiply the remaining decimal by 60. This
is the number of seconds.
(.54)(60)=32.4
32.4”
Convert decimal degrees to DMS.
1. 20.5°
2. 46.327°
3. -189.62°
Let θ = 56°34’53”. Find the complement and
supplement to θ.
Let θ = 56°34’53”. Convert θ to radians.
Objectives:
1. To find linear and
angular speed
2. To find the area of a
sector
3. To convert between
DMS and decimal
degrees and use
DMS in angle
calculations
Assignment:
• P. 292: 91-94 S
• P. 292-3: 101, 103, 105,
106, 108, 113, 116
• P. 291: 71-74 S
– Also find a complement
and supplement if
possible
– Also convert to radians
• P. 291: 75-78 S