Angle Measurement With Geometrical Concept
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Transcript of Angle Measurement With Geometrical Concept
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An angle is formed by joining the endpoints
of two half-lines called rays.
The side you measure from is called the initial side.
Initial Side
The side you measure to is called the terminal side.
This is a
counterclockwiserotation.
This is a
clockwise
rotation.
Angles measured counterclockwise are given a
positive sign and angles measured clockwise are
given a negative sign.
Positive Angle
Negative Angle
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Its Greek To Me!It is customary to use small letters in the Greek alphabet
to symbolize angle measurement.
E F
H
K
JU
alpha beta gamma
theta
phidelta
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We will be using two different units of measure
when talking about angles: Degrees and Radians
Lets talk about degrees first. You are probably already
somewhat familiar with degrees.
Ifwe start with the initial side and go all ofthe way around in a counterclockwise
direction we have 360 degrees
You are probably already
familiar with a right anglethat measures 1/4 of the
way around or 90
U = 90
Ifwe went 1/4 of the way in
a clockwise direction the
angle would measure -90
U = - 90
U = 360
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U = 45
What is the measure of this angle?
You could measure in the positive
direction
U = - 360 + 45You could measure in the positive
direction and go around another rotation
which would be another 360
U = 360 + 45 = 405
You could measure in the negativedirection
There are many ways to express the given angle.
Whichever way you express it, it is still a Quadrant I
angle since the terminal side is in Quadrant I.
U = - 315
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If the angle is not exactly to the next degree it can be
expressed as a decimal (most common in math) or in
degrees, minutes and seconds (common in surveying
and some navigation).
1 degree = 60 minutes 1 minute = 60 seconds
U = 2548'30"degrees
minutesseconds
To convert to decimal form use conversion fractions.
These are fractions where the numerator= denominator
but two different units. Put unit on top you want toconvert to and put unit on bottom you want to get rid of.
Let's convert the
seconds to
minutes
30"
"60
'1 =0.5'
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1 degree = 60 minutes 1 minute = 60 seconds
U=
2548'30"
Now let's use another conversion fraction to get rid of
minutes.
48.5'
'60
1r = .808
=2548
.5'=
25.808
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initial side
radius of circle is r
r
r
arc length is
also rr
This angle measures
1 radian
Given a circle of radiusr
with the vertex of an angle as thecenter of the circle, if the arc length formed by intercepting
the circle with the sides of the angle is the same length as
the radius r, the angle measures one radian.
Another way to measure angles is using what is called
radians.
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Arc length s of a circle is found with the following formula:
arc length radius measure of angle
IMPORTANT: ANGLE
MEASUREMUST BEINRADIANS TO USE FORMULA!
Find the arc length if we have a circle with a radius of3meters and central angle of0.52 radian.
3
U = 0.52
arc length to find is in black
s =rU3 0.52 = 1.56 m
What if we have the measure of the angle in degrees? We
can't use the formula until we convert to radians, but how?
s =rU
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We need a conversion from degrees to radians. We
could use a conversion fraction if we knew how many
degrees equaled how many radians.
Let's start with
the arc length
formula
s =rUIf we look at one revolution
around the circle, the arc
length would be the
circumference. Recall that
circumference of a circle is
2Tr2Tr=rU
cancel the r's
This tells us that the
radian measure all theway around is 2T. All the
way around in degrees is
360.
2
T=
U2 T radians = 360
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2 T radians = 360
Convert 30 to radians using a conversion fraction.
30r
360
radians2T The fraction can bereduced by 2. This
would be a simpler
conversion fraction.180
T radians = 180
Can leave with T or use T button on
your calculator for decimal.
Convert T/3 radians to degrees using a conversion fraction.
radians
180radians
3 T
T r
= radians6
T= 0.52
= 60
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Area of a Sector ofa Circle
The formula for the area of asector of a circle (shown in
red here) is derived in your
textbook. It is:
U2
2
1rA !
Ur
Again U must be in RADIANSso if it is in degrees you must
convert to radians to use the
formula.
Find the area of the sector if the radius is 3 feet and U = 50
rr
180
radians50
T=0.873 radians
213 0.873
2
3.93 sq ft
A
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radians
inbemustU
speedangular
theis[
When something
is rotating we can
measure how fast the
angle is changing called
angular speed. This is
generally designated by
the Greek letter[.
t
U!
timeist
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We can also compute linear speed of an object that
is rotating. Think of placing a mark on a wheel and
then keeping track of how far the mark travels. Itwould travel around the circle so it makes sense to
use arc length s for the distance.
The mark traveled the distance ofrU (since s =rU)
t
sv !speedlinearspeed = d
istance
time
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t
U!
t
sv !
So we have angular speed [and
linear speed v. We can get a
another formula for linear speedby doing some substituting.
t
s
v !
s =rU
t
r
v
U!
t[U ! This is angularspeed formula butsolved for U
t
tr
v
[
! [rv !
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t
U!
t
sv !rv !
Let's try a problem: The radius o each wheel o a car is 15 inches.
I the wheels are turning at the rate o 3 revolutions per second, how
ast is the car moving?
We want linear
speed so we'll use[rv ! To ind [we'll use theangular speed ormula
t
U
!
We need radians per time but we have revolutions per
second so we'll convert that to radians per second.
sec
rev3
rev1
radians2T
conversion raction since 1 revolution is 2T radians
sec
radians6T!
sec
in615 T!v
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sec
in90T!v
Since we don't have a feel for how
fast inches per second is in a car,
let'
s convert this to miles per hourusing conversion fractions and the
pi button on the calculator.
sec
in7.282!v in12
ft1
feet5280mile1
min1sec60
hour1min60
hour
miles1.16!v
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A Sense ofAngle Sizes
See if you can guess
the size of theseangles first in degrees
and then in radians.r45
4T! r60
3T!
r30 6T
!
r1203
2T!
r180
T!
r902
T! r150
6
5T! r135
4
3T!
You will be working so much with these angles, you
should know them in both degrees and radians.