Angle Measurement With Geometrical Concept

8/6/2019 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.

Angle Measurement With Geometrical Concept

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