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in this episode you’ll learn about measuring trigonometric angles in both

radians and degrees we’ll also study arc length and the area

of a sector let’s do some math hi everyone welcome back we’re going to

begin with the question what are trigonometric angles so

um you you we’re going to have a vertex and we’re going to have an initial side

and we’re going to have a terminal side and the angle here isn’t going to be an

opening between two rays the angle is going to be the amount of rotation

so this is going to be the amount of rotation and that’s the angle

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this is the terminal side so we have a vertex let’s call that vertex here

and we have a terminal side an initial side and how do we know the terminal

side and the initial side well the arrow’s pointing towards the terminal

side in other words we’re going to imagine that the terminal side right

here started with the initial side and the two rays were on top of each other

and then we’re going to take the terminal side and rotate around the

vertex here and as we go around and then

when the terminal side stops now we have an opening we’re going to call that a

trigonometric angle trigonometric angles measure the amount of rotation so this

could go around multiple times so this angle right here would be different than

this angle right here that’s a lot more rotation we went around all the way

and then we went some more um and then this angle would also be

different so here’s three different angles right here that we’re drawing

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in fact this is an example of a negative angle because the direction of rotation

this is a positive angle and this is these are two different positive angles

we can go around as many times as we want we can go around this way as many

times as we want so trigonometric angles measure the rotation

and we have something called the standard rotation standard uh standard position

so standard position is when the initial side right here the vertex is centered

at the origin right here and then we have the

initial side which is along the x-axis and then the terminal side can be

wherever and so i’ll just draw it right over here for example

and so this would be some amount of rotation

and we often label uh angles with greek letters uh a popular one is theta so

i’ll just write that as theta right there so this is the amount of rotation

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so imagine that being right on top of the x-axis right there and

i’ll just redraw it so that it is so it’s right on top of it

and so there’s the initial side here’s the terminal side right here and the

angle is the amount of rotation right there so this is um an angle in standard

position right here vertex is at the origin initial side and then the

terminal side is somewhere um and so if we’re going to be measuring

counterclockwise we’re going to have positive angles and if we’re measuring

clockwise then we’re going to have negative angles so for example this

angle right here that angle right there this one was i called it theta this

thing right here is minus 90 degrees now i haven’t said what 90 degrees is yet

we’ll talk about degrees here in a moment so let’s just call it um you know this

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is a quarter of a revolution but it’s a minus because we’re going uh

you know clockwise um coterminal angles so coterminal

angles are angles that are have the same initial side and the same terminal side

so let me draw a couple of examples so let’s say this is the initial side right

here initial side of the angle and let’s say the terminal side was

let’s say right here so this is the terminal side so i’ll just draw it with

the angle we don’t need to label it with words

um initial side terminal side right there and so this is a

you know angle right here that’s uh vertex is at the origin

and i can say that this right these two angles right here are coterminal

so these are coterminal because they have the same coterminal angles

these two angles are coterminal and the reason why is because they have the same

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initial side which is this ray right here and they have the same terminal

side which is this ray right here so it doesn’t matter how they’re

oriented positive or negative coterminal

just means they end up at the same place but also they started at the same place

and then let’s um talk about what a radian is now so what is a radiant

so radian is we’re going to um first of all we’re

going to have a circle so let’s draw a circle let’s suppose that’s a circle

and here’s the center about right there and what we’re going to have here is a

central angle and so i’m going to just draw this up here and

let’s call this right here r and this distance right here this curved

distance right here we’re going to call that s

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and we’re going to call this right here theta this angle right here is in theta

right here so you can imagine that this is the initial side and then we’ve

rotated and we’ve got a terminal side right here

and you know so this is going to be a central angle central angle theta

and what we’re going to have is that s equals r theta s equals r theta

and so now let’s think about this in terms of units for a minute s is a

unit of measurement of length and so is r

in fact if you want you can put s over r

the point is is that if s is measured in inches and and r is measured in inches

the inches cancel out and this theta here is unitless

um and so you know that’s important so theta is going to be measured in radians

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so what is radians so whenever we have um s equal to r whenever this

this arc length right here is exactly the same as the radius

then that will determine the angle and that angle is going to be called one

radian so one radian is when s equal to r so just a quick diagram over here

we got this r and this r and we got this r

these are both r’s because they’re both the radius but this angle in here is

theta and you know we have theta equals s over r but that but the s is

r so this will be one radians one radian and so um

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now what we’re going to do is um you know just want to make sure that you

understand that the trigonometric angle measures the amount of rotation

and what one radian is right so we have the radius and the arc length are the

same of a circle and that angle right there is going to be exactly one radian

and so let’s look at a couple of examples so what is a half of a revolution

a half of a revolution one half of a revolution i’ll just abbreviate revolution

um and so this will be two pi over two which is pi radians so

or or let’s start off with say one revolution is two pi radians

so one full turn around the circle is two pi radians

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and so that will tell us that the you know the circumference of the circle is

right the circumference just in case you’re not not remembering the

circumferences 2 pi r of a circle and so you know when the circumference is

the arc length here then we can determine that one revolution is two pi

radians right there and so if you have a fourth revolution

then it’s just a simple matter of dividing by four

right and so then we just get pi over two radians and when we divide by 6

so these are some common numbers here if we divide by 6 we get pi over 3 radians

and as we move along we’ll we’ll we’ll name some special angles

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but definitely one revolution is two pi half revolution is pi

and this one is the one we’ll use to convert degrees to radians we’ll we’ll

talk about that in a minute of course you could have a quarter of

revolution which will be pi over two radians um

and so just one more thing before we go just to make sure everyone’s on the same

page here we’re going to call this one here quadrant one quadrant two

quadrant three in quadrant four and the reason why this is important is because

when we measure angles remember angles measure rotation so i can take the this

initial side right here and i can rotate around and i can go

around multiple times as many times as i want because this rotation how many

times you’re going to rotate around and then i can end in one of the

quadrants let’s say i end in quadrant three and so this right here would be the

angle right there and we can call that angle theta and that angle is um [Music]

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you know a positive angle i drew it uh you know going going

counterclockwise so that’s a positive angle right there

um but that angle theta is large much larger than just say

if i have the same ray right here then this one right here so they both

end up in quadrant three down here but they’re not the exact same angle

right this angle is much larger than this one this one is you know less than

360 degrees or it’s less than two pi this one went around multiple times

so you know that’s important there to realize

we can talk about them as both being quadrant three angles but this one has

uh this one is much larger has much more rotation

all right so how to sketch and find coterminal angles um so here we go

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let’s get rid of this real quick okay so um let’s find the coterminal angles for

um let’s look at pi over six first so let’s just sketch the graph for pi over

six right here and so let’s say pi over six is about right here

let’s call this here angle right here is pi over six

and right so two pi is all the way around and pi over two is up here

so we got pi over six hanging up hanging off about right here

now if you’re not good with uh figuring out where these angles are

um it doesn’t hurt to actually go around the circle

just to kind of doodle for yourself and think about that um

so let’s just draw a quick circle around here we’ll get back to this here in a

second here let’s look at just going around the circle right here

so we know going around the circle is two pi that’s one revolution and one and

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a half revolution is pi so this is pi right here um so if you go two pi

think about things in terms of six and this is pi right here

think about this in terms of six right so this will be pi over six

so that angle right there is pi over six and this will be two pi over sixes

two pi over six and then this right here will be this angle right here will be

three pi over six and this one right here will be four pi over six

and this one here will be 5 pi over 6 and this pi right here you can think

about it as 6 pi over 6 right which is just pi

and then this is seven pi over six and this one coming right through here

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will be eight pi over six now eight pi over six reduces to four pi over three

so that was eight pi over six so this will be nine pi over six

you know three pi over six that reduces down to pi over two

two four pi over six r uh reduces down to two pi over three

just reducing the fractions 9 pi over 6 reduces to 3 pi over 2

but don’t forget that was 9 pi over 6 so the next one coming right through here

like this right here is 10 pi over 6 and then the last one coming right

through here will be 11 pi over 6 because this 2 pi will be the 12 pi over 6.

12 pi over 6 reduces to 2 pi and this one was 11 pi over 6 and then

we should reduce the 10 pi over 6 to 5 pi over 3.

and so that’s just measuring the positive angles right there

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yeah so just to kind of doodle a little bit so that you can kind of get a

feeling for these angles right here in any case coterminal

what are some coterminal angles right here so we’re going to say here

that this angle right here is coterminal we can go all the way around and then

come right back and so let’s say pi over 6 plus 2 pi

and so let’s just say that’s pi over 6 and then 2 pi we’ll just say it’s 12 pi

over 6 so we’re going to get 13 pi over 6 so this angle

so these two angles are coterminal pi over six and thirteen pi over six

now we could find a lot more angles that are coterminal we could just add keep

adding two pi over two pi’s and we’ll keep getting more and more

coterminal angles um what about 13 pi over 6 so where’s 13 pi over 6

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well that’s the same one right here right so we can find more

another angle that’s coterminal with 13 pi over six so we already know these

two are coterminal but we can do this again 13 pi over six

and then plus another two pi to to rotate all the way around again so we’re

going to go uh 13 pi over six and then we’re going to

go around two more pi we can put that say in red or something like that

so this will be you know and then we’ll go around two more pi

so plus two pi and this will be 13 pi over 6 plus 12 pi over 6

and adding those up we get 25 pi over six so 13 pi over 6 and 25 pi over 6 those

are coterminal also so what about if we have minus two pi over

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three so let’s draw this one over here so positive two pi over three if you

remember uh a second ago we were doodling 2 pi over 3 was over here

but what about negative 2 pi over 3 where is that at

right so now we have to go back down this way so negative 2 pi over 3

and let’s just erase that so negative 2 pi over 3 is about right here

negative 2 pi over 3 and i’ll just you know label those

but you know negative two pi over three and so now we’re looking for uh angle

that’s coterminal now one way to do it would just be to add two pi to it i can

say minus two pi over three and then do another minus two pi

and we’ll certainly get a coterminal angle so minus two pi over three

and then minus two pi and so that’ll be minus two pi over three

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minus uh 12 pi over 6 right and so you know we could we could

do that that’s no big deal but actually i want to find this angle

right here let’s put it in red these two angles are also coterminal

and this angle right here is positive and it’s nice to have positive angles so

i want to find not only the coterminal angle but i want to find a positive

coterminal angle for no apparent reason so i could this one this angle right

here will be coterminal just put the fractions together but i actually want

to do pi plus a little bit more right so this

angle right here in red is going to go all the way to pi

half of a revolution and then it’s going to go a little bit more

so how much more did it go so you know this right here is um you know

minus pi over three right here or oops sorry um so we need to

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you know make sure that these angles add up to

two pi right here we’re gonna go here and then here it’s gonna be one full

revolution right here so pi plus and then a little bit more right here

for trying to find this angle right here pi plus a little bit

and then plus two pi over three and that should all be equal to two pi

in other words going all the way around so i got that from half of it pi

and then plus a little bit more here and then this this angle right here will

be the 2 pi over 3 and that will give us a full revolution

two pi right there so what will be this angle right here so we can think about

this as this pi right here as three pi over three or or let’s just

solve for theta right here so this will be two pi

and then move the minus 2 pi over 3 over and then move the minus pi over

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right and so just you know say 2 pi and then move

this one over it’s minus move this pi over it’s minus so put all these together

and then the uh angle that we’re looking for will be pi plus this right here

in fact let’s just actually keep the pi over here

pi plus because this is really the coterminal angle that we’re looking for

right here so think about like this 2 pi which we’ll say is 12 pi

or to get a common denominator here let’s just say 6 pi over three

minus two pi over three which is four pi over three right

uh let me see if i can move out of the way so this will be four pi over three so

this angle right here is 4 pi over 3 and you know if we do 4 pi over 3 plus the

you know 2 pi over 3 that will give us 6 pi over 3 which is 2 pi

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so anyways that’s how i got this angle right here in red it’s 4 pi over 3

and this is coterminal another way to find a coterminal angle

would have been to uh you know just have found this angle right here um

but then have gone around one more time so there’s two ways

to do this this will be the first way this will be the second way

so anyways finding coterminal angles there are some examples there

and now let’s talk about complementary and supplementary angles

all right so if possible find the complementary and supplementary angles

and the first one we’re going to look at is 2 pi over 5.

so we we’re going to need to remember what complement complement means you add

them up and you get 90 degrees and supplement means you add them up and you

get to 180 degrees or so differently because

we’ve already talked about radians we want to add them up right so we want to

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say uh you know alpha plus beta equals pi over two

so that’s what complementary means when you add them up you get the pi over two

uh a quarter of a revolution and supplementary means you take your

two angles and you add them up and you get 180 or you get pi pi radians

so pi over 2 radians or you get pi so let’s look at this one right here

first two pi over five so i’ll call this one part a two pi over five

so um i’m going to say here that this is going to be

alpha plus two pi over five needs to be pi over two

so in other words this angle that we’re looking for the complementary when i add

these two together i have to get pi over two

so we can solve for the missing angle it’ll be pi over two minus

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uh two pi over five and now just getting a common denominator right so you know

multiply this will be five pi over 10 and this is a minus

and this will be 4 pi over 10 and then you know 5 pi minus 4 pi is

just 1 pi so pi over 10 and so 2 pi over 5 and pi over 10

these are complementary angles right here so complementary and

to be supple supplementary so we’re going to say

uh let’s let’s just use a different um angle let’s just use beta

plus 2 pi over 5 needs to be the 180 degrees or in other

words half of a revolution which is pi radians so

to solve for this beta i’m going to say beta is pi minus 2 pi over 5

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and now for this pi we’re going to use 5 pi over 5 minus 2 pi over 5

which is going to be 3 pi over 5. and so this is a supplement

supplement angle so in other words if you take these angles together

you add them up and you get pi all right so now let’s do the this

second one here 4 pi over 5. let’s look at that one right there

let me get rid of this real quick all right so part b here 4 pi over 5.

okay so we need to add them up and get to 90 degrees

so i’m going to say alpha plus 4 pi over 5 is going to be pi over 2.

all right and so alpha’s going to be pi over 2 minus 4 pi over 5

and so let’s get our common denominator again

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so let’s say this is be 5 pi over 10 and then 8 pi over 10

and as we can see here this is a negative angle right here

so when we add these up we’re we’re going to get pi over 2 but

when you say complementary angles we also need the angles to be positive

right so no complementary no complement so in other words this angle is already

too big to have a complement its complement if it existed would have to

be negative right so we’re going to say no complement here there’s an m in there

somewhere complement all right and so now let’s go with the supplement

so alpha plus 4 pi over 5 now or let’s use the beta so now let’s

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say this has to be pi so beta is pi minus four pi over five

and so let’s say that’s five pi over five minus four pi over five in other

words pi over five and that is the supplement okay so i hope that helps you

refresh your memory on complementary and supplementary angles

and now let’s talk about degrees so what are degrees so um

we’re going to say that a second way to measure angles degrees

is a way to measure angles so we’re also going to use amount of rotation

and we’re going to use the symbol that’s like a little circle sign

so we’re going to say that 1 degree is equal to 1 over 360 revolutions

one over 360 revolutions and so what we’re going to need to have

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is a way to talk about degrees and radians together so if i multiply here by 360

so we’re going to get 360 degrees is 2 pi radians 2 pi radians and that’s

one complete revolution and 180 degrees where half of that is pi radians

and this is a half of a revolution and so

we’re gonna get that one degree is also pi over 180 degrees radians

and sometimes we’ll just abbreviate rad and one radian is 180 degrees over pi

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and so these are our conversion rules here if you want to go from degrees to

radians or radians to degrees we can use these conversions right here and so

let’s just make another doodle but now let’s put in here the pi’s and the

you know the radians and the degrees let’s do that real quick

so let’s see if we can get a decent circle right here i’ll use my um coaster

all right that’s good enough um and let’s see if we can pick the

center right there and let’s see if we can get

a straight edge here just to make it look decent and

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let’s put on here the radians in degrees here all right and so let’s um you know

cut this in force first of all so this will be 90 degrees or pi over two

i did half of a of a half and this will be 180 degrees or pi pi radians

and so this will be 1 pi over 2 2 pi over 2 three pi over two [Music]

so three pi over two or we can do 90 plus 90 plus 90 which will be 270 degrees

and then we can do the last one right here which is zero degrees or zero radians

or that’s if you didn’t go anywhere or if

you went one time around it would be two pi radians and 360 degrees

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if you want to measure the angle right there like that all right and so

i messed up my circle there so now let’s go around so we chopped everything

into quarters now let’s do everything into sixths

so i’m going to say the first one here is pi over six which if we convert this

how do we convert pi over six to degrees so i’m going to do 180 degrees over pi

remember those are the same right there but this will cancel the pi’s this will

leave us with degrees so what’s 180 over 6 this will be 30 degrees

and so this will also be 30 degrees right here so this angle right here

is pi over 6 radians or 30 degrees whichever way you want to think about it um

now before we do that though before we continue with those

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let’s do the 45 so i already chopped the quarters out let’s do that again

i want to take the 45 degrees out how do you convert 45 degrees to radians

oops 45 degrees to radians so now i’m going to say pi over 180 degrees

because i want degrees to cancel and get radians

so i’m going to end up with a pi but what is 45 over 180 right so it can

go in there four times right so double that you get 90 and then

double that right so pi over four this is pi over four right here

so this angle right here is pi over four in other words it’s half of pi over two

it’s 90 degrees half of 90 degrees here and so now i want to do pi over four is

all the way around so here’s pi over four and then here’s two pi over four

and then here’s three pi over four so let’s go right through here right

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through the middle right there three pi over four and what is that in degrees

well one way is to add 45 add another 45 and then add another 45

or i can just simply convert it 3 pi over 4

and then 180 degrees over pie that will cancel the pies so four goes into 180 um

we’re going to get 135 degrees right so let’s just put that up here 135 degrees

and then we’ll go another 45 degrees right here and we’ll get 180 and we’ll

go another pi over four so we want 1 pi over 4 2 pi over 4 3 pi over 4

4 pi over 4 4 pi over 4 reduces and then we’ll have 5 pi over 4

and then we’ll come through here again we’ll have 6 pi over 4 which reduces so

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that’s six pi over four that reduces and then this one will be

seven pi over four and we can convert seven pi over four uh 180 over pi

but it’s probably just as easy to count up to 45 degrees 145 degrees 245 degrees

345 degrees 445 degrees 545 degrees 645 degrees 745 degrees so what’s 7 times 45

35 right and so then we get here you know 315

if we had 45 more then we get back to the 360.

all right and so that’s doing the pi pi over fours

but let’s also do the pi over sixes so the next one will be two pi over six

which will reduce down to pi over three which if you convert it to degrees is

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sixty degrees and i know that because this will be one thirty degrees this

will be two thirty degrees and then this will be 330 degrees in

other words 90 degrees and so i can go another 30 degrees right here

and so another 30 degrees will be 120 and what will be in radians so 1 pi over

6 2 pi over 6 3 pi over 6 so this would be 4 pi over 6

but 4 pi over 6 reduces to 2 pi over 3 so this will be 2 pi over 3 right here

all right and we can keep continuing this in all the quadrants

1 pi over 6 1 pi over 6 2 pi over 6 3 pi over 6 4 pi over 6

this right here will be five pi over six this will be six pi over six

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so i get pi this will be seven pi over six this will be eight pi over six

this will be nine pi over six this will be ten pi over six

and this last one here will be eleven pi over six

and then we can go and we can reduce each of these 11 pi over 6 is reduced 10

pi over 6 is 5 pi over 3 [Music] 8 pi over 6 is 4 pi over 3’s

seven pi over six is reduced six pi over six is reduced

five pi over six is reduced two pi over three is reduced

um and then now we can go and get the degrees 30 degrees 230 degrees 330 degrees

430 degrees and so this will be 530 degrees which is 150

and this will be 630 degrees and this will be 730 degrees so that’ll be 210

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and this will be 830 degrees which would be 240

and this will be 930 degrees so that’ll be 270 right there

so that’ll be nine so this will be 10 30 degrees 300

and then the last one will be 11 30 degrees or 330 degrees right there

so later on we’ll add more stuff to this um but this just gives us the angles um

and so i feel like that kind of like helps you

get much more comfortable with you know what degrees are but we defined one

degree as being one over 360 of a revolution okay so um

now let’s practice a little bit more converting between degrees and radians

where you just can’t count them up like that

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so let me get rid of this real quick right here so let’s convert

let’s convert seven pi over nine so seven pi over nine wasn’t on our circle

uh that we just laid out so how do we convert so you either gonna

use pi over 180 degrees or 180 over pi i recommend using one of those two now

you don’t have to you could use two pi over

360 degrees or 360 degrees over two pi you could use either one of those to

convert or any of these four to convert but you know why make things more

complicated than you need to you know you could even use four pi over

720 degrees you know just doubling everything right

so it’s usually easier to do these right here to do whichever one you need using

any one you need this is already in radians

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so i want to convert to degrees right we want to convert this to degrees so we

want degrees on top so i’m going to use uh

and that’s a 7 by the way so 7 pi over 9 so 7 pi over 9

times 180 degrees because that’s what i want degrees divided by pi

so now the pies will cancel and i get 7 times 180 over 9

and then that’ll be degrees and then you know so 7 pi over 9

this gives us 14 degrees if we reduce that right there

all right and so now let’s convert 10 degrees to radians

so now i’m going to have 10 degrees again 10 degrees wasn’t on our

circle that we just doodled a minute ago with all the special angles on it

we didn’t go around every 10 degrees we went around every

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30 degrees and every 45 degrees we could have gone around every 10 degrees if we

wanted to but you know we didn’t and so let’s just convert it right off

so 10 degrees and we’re going to say we’re going to need 10 times 10 degrees

times and now i want degrees to be down here so they cancel someone have pi

and so we’re going to get you know pi over 18 radians

um okay and so you know let’s do two more

so let’s convert here the 45 degrees to rate 450 degrees to radians

so 450 degrees and this is going to be 450 degrees times pi over 180 degrees

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and we’re going to get 5 pi over 2 radians now

this is 5 pi over 2. now this is the exact value if you wanted to you could

go to your calculator and you can get a decimal and you’ll get something like 7

785 but i just want to take a moment to mention that 785 is an approximation

you can never write all the decimals down for pi

so you might your calculator might use three decimals eight decimals 12

decimals who knows what your calculator or computer program will use when when

approximating pi so if you’re going to use 7.85 make sure

you use the correct symbol you should use the approximation symbol here

because this is an approximation this is an exact value so in other words if the

problem asks us to approximate 450 degrees in radians then

this would be an acceptable answer but it doesn’t say that it says convert

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so this right here 5 pi over 2 is the correct answer um and then this is just an

approximation but this is has some use if you’re trying to plot this or you

know look at this further than knowing what it’s approximately equal to can be

helpful but technically this right here is the answer and so i’m just going to

erase that right there all right one more how about convert pi

over six to degrees right and so we already practiced that right here so pi

over six we’re just going to say you know that is going to be equal to

180 degrees over pi pi’s cancel and we’re going to get 30

degrees and so for this right here i just want to mention that

if you leave off your degree symbol just by accident then

you know when there’s no no unit written the default value is radians so 30 is

the same thing as 30 radians so some people will put radians some

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people just have understood that if you don’t have anything then that’s radians

right there so this is supposed to be degrees this is degrees here so i just

want to mention that if you leave off your degree symbol that could have um

important um you know communication problems all right so how do we find the arc

length here let’s look at arc length now we’re going to look at two applications

of this radiance and degrees thing here so so recall that we had here

a circle and i’ll just put one right here

and we this is the radius of the circle and we have some

arc length here which we labeled as s and we have some angle here which we call

theta and the relationship we’re talking about here is theta is s over r

or said differently um s equals r theta and what’s important to realize here

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is you know what i was just talking about second ago is the units here are

measured in length and so the units actually here cancel and so theta here

you can think about it as being unitless or you can think about it as in radians

all right and so this will be the um arc length formula right here um and so

you know s equals r theta this will give us the arc length here if you know the

radius of the circle and you know the how far of an angle that you went

through here let’s call this here angle theta right here

and then you have s equals r theta but you know you have to remember that this

theta right here is in radians so theta is in radians all right so

let’s look at an example so let’s say here a circle has radius four inches

so we’re going to say r is four and we’re going to use a central angle

00:44

right here of 240 degrees so theta is 240 degrees

now we cannot rush in here and say oh i got the r i got the theta because

remember this theta is in radians right here so we cannot go say s is equal to 4

times 240 so we’re going to need to know that we need to convert this degrees

they gave us degrees so we’re going to have to convert this to radians

so let’s convert this to radians so 240 times pi over 180 degrees

and so this will be 4 pi over 3 radians so i’ll just put 4 pi over 3 here

and so now we can use s equals r theta so s is equal to 4 they give us here 4

and and this is this 4 here is in inches so this would be 4 times uh four times

the four pi over three and so this gives us 16 pi over three

00:45

and then we should put on of course the inches here so 4 pi over 3 inches

or just let’s just say i in here inches all right and so you know that’s the

length of arc so if this was 4 and this theta right here was 240

degrees which this is drawing here isn’t accurate for 240 204 degrees we can put

one over here real quick suppose that’s a circle

and then you know there’s the center 240 degrees is you know not quite 270

so there’s our angle theta right here and we’re going to ask you know what is

this arc length here you know this distance right here let’s put it in red

and so we’re we’re knowing that the radius is four here so from here to here

is four or you could say from here to here is four

00:46

but the angle right here is 240 degrees or as we found it in radians 4 pi over 3

radians and so that was the angle theta and this this distance right here s

is 16 pi over 3. now that’s the exact value you may go to a calculator and get

approximate value which is 16.76 and then the same units here of course 16.76

so that gives us some kind of um you know approximation i think that that

has a certain value to it because the this was 4 and this was

16.76 so that kind of gives us some perspective rather than just having the

exact value 16 pi over 3. in any case let’s look at

another example right here let’s look at two more real quick

00:47

so this one right here i’m going to um so let’s call this one here example a

we’re going to find the arc length of the circle and the radius is 10 meters

and we’re going to have a central angle of 30 degrees

so this will be our central angle and so this is the length and so we want

to find s find arc length s and so we can say

we’re going to convert this first here so pi is also 30 degrees we converted

that as pi over six and so the arc length s here is ten times pi over six

and so just reducing that to five pi over three

and then we would just add in our unit of measurement remember arc length

00:48

that’s length this was given in meters so that’s really 10 meters times a number

so this will be meters for part b here um we’re going to say a central angle

theta uh so i didn’t draw this one here but

just to make sure that you’re okay on central angle there i’ll just draw it

right here real quick just have a circle right there and then this is

um 30 degrees right here and this is the angle and this is the 10 meters right

here and so you know this angle right here is its

central angle in other words that goes to the center of the circle right there

and the initial side is along the x-axis there any case

so on this example here b we have a central angle

and we have a radius of 4 meters and an arc length of 6 meters

00:49

right and so we want to find theta find theta in radians in radians

right and so you know we have three we have s equals r theta if we’re given any

two we can find the third right so in this example here

you know r is four s is six and so theta is just you know divide by

four so it’s just three over two or you know 3 over 2 radians

so really nothing more to it than that but just keep this in mind that this is

in radians and r and s have the same unit of

measurement to make the equation equal but given any two you can find the third

all right so now let’s talk about the area of a sector so let’s do that now

00:50

so uh don’t forget that this uh episode is part of the series

uh trigonometry is fun step-by-step tutorials for beginners

and so check out the link below in the description

and you can see the other episodes all right and so let’s see how to find

the arab sector now arabic sector arabic sector of a circular

area of a sector of a circular uh that’s wrong let’s word that correctly here

how do you find the area of a circular sector formula area of a circular sector

and so let’s see if we can get that fixed here

area of a circular sector formula all right there we go

and then we’ll look at an example here in a second all right so let’s draw a

00:51

circle again i’ll just kind of eyeball one here um

and let’s not make it go through that word let’s put another one right here

all right and let’s say here we have some it doesn’t have to be a central angle

um let’s call it angle theta right here central angle right there

goes through the center and we got r right here some radius of the circle

right there and so we’re going to have the area of this part right here in red

and this right here is going to have area is going to be one half

r squared theta and this a theta here is measured in in radians in radians

now if you forget that kind of think about it making sense here

00:52

remember area is going to be for example inches squared or centimeters squared

right and so when you square the radius that’s what’s going to happen if r is

say 10 inches then this is going to be inches squared

and that’s going to be the area so in other words theta isn’t going to be

measured in degrees or any other kind of units it’s going to be unitless or it’s

going to be in radians say the same thing right so um you know

this r is the radius and a is the area in red here of this of the circular

sector okay so let’s do an example now so let’s say a sprinkler on a golf course

um 70 feet for our radius and we have 120 degrees for our

for our angle right here so the first thing is we’re going to have to convert

00:53

this 120 into radians so let’s say 120 times you know pi over 180

so the degrees will cancel so we end up with 120

over 180 and so we’re going to get 2 pi over 3 and then this won’t have any more

degrees we’ll just call it radians and so now we can use this theta right here

in our formula to find the area of the watered uh you know area

so the area will be one half and then the radius was 70 feet

and then times oh and then squared and then times the angle right there

which is two pi over three and so then we can just you know

manipulate this and find this value right here

so we’re going to get 4 900 over 3 pi you know just squaring that

and then the 2’s cancel and you get over 3 and then this will be pi

00:54

and then you know because this right here is um in terms of pi

you know actually the pi and the three here is kind of nice

because the you know you get 0.14 blah blah left over

so you know you can kind of know that these two numbers are the same

are close to each other so it’s going to be roughly 4 900 feet but actually if

you approximate it it comes out to be 5 131 feet and let’s put the units on and

let’s say here square feet and so we’ll just say feet squared here

so approximation but here’s the exact value right here so it’s roughly

4900 but you know you can get a lot better than that right there

by using a calculator all right and so uh let’s look at one

more here real quick um just you know real briefly here

00:55

um find the area of a sector of a circle uh central angle 60 degrees and the

radius is three so i’ll just sketch it real quick and we

got 60 degrees right here and we’re looking for the area in here radius is 3.

so the area will be one half times the radius squared times the angle

and so we need to know what 60 degrees is in radians right

so that was uh pi over three and so you can convert 60 degrees to radians but

it’s one of the ones we did before and so now we can just you know simplify

this right here so this will be just you know three pi over two

00:56

and this is meters squared and so that’s it hope you enjoyed this video

and uh i’ll see you in the next episode have a great day if you enjoyed this

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