Radians and Degrees Explained (Step-by-Step)

Video Series: Trigonometry is Fun (Step-by-step Tutorials for Beginners)

(D4M) — Here is the video transcript for this video.

00:00
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

00:01
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

00:02
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

00:03
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

00:04
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

00:05
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

00:06
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

00:07
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

00:08
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

00:09
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

00:10
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]

00:11
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

00:12
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

00:13
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

00:14
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

00:15
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

00:16
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

00:17
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

00:18
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

00:19
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

00:20
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

00:21
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

00:22
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

00:23
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

00:24
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

00:25
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

00:26
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

00:27
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

00:28
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

00:29
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

00:30
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

00:31
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

00:32
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

00:33
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

00:34
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

00:35
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

00:36
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

00:37
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

00:38
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

00:39
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

00:40
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

00:41
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

00:42
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

00:43
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
video please like and subscribe to my channel
and click the bell icon to get new video updates

About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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