The Unit Circle With Special Angles (Constructing the Unit Circle)

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 more about the unit circle we’ll construct a unit
circle with special angles and we’ll also go over the concept of reference
angles then we’ll explain what all this has to
do with triangles let’s do some math [Music]
hi everyone welcome back we’re going to begin with this question here
what is a unit circle so surely you’ve heard of the word unit
circle especially if you’re starting out studying trigonometry
and so today we’re going to go over exactly what it is
now before we get started though i wanted to mention that this episode is
part of the series uh trigonometry is fun step-by-step
tutorials for beginners the link is below in the description
and i think this is um the second episode in the uh series or it’s one of

00:01
the first ones in the last episode we talked about what are trigonometric angles
and we talked about uh moving from degrees to radians
and so i wanted to go over the unit circle with this today
so um first of all what is the unit circle
so unit circle is basically or is just exactly the uh a circle
the circle with a radius one and so that’s why we call it a unit
circle radius is one and so what’s an equation well an equation for the unit
circle is if you um set the circle on the origin as the center of the circle
then a nice simple equation is x squared plus y squared equals 1 squared
right and so this is the unit circle centered at the origin so unit circle
um and you might call this standard position but we usually reserve that for

00:02
angles so i’ll just say unit circle centered at the origin
and you know i’ll draw a picture of this here in a second because um the third
one here is you know the set of points right so let’s draw a nice circle
and then we’ll put down the x and y axis here’s an equation for it
and that right there is the center so the radius is one so this right here
is the point one zero and this is the point right here zero minus one
and this is the point right here minus one zero and this is the point up
here which is just 0 1 right and so there’s a there’s you know
a couple points on the unit circle there
now notice that there’s lots of symmetry though so if i take any point over here
call this point here say x y then what’s the corresponding point over here

00:03
right so this is symmetric about the y axis so this point right here would be x
and then okay so it would be minus x and then it would have the same y
so whatever this y is would have the same y but this would be
a positive x and this would be a negative x
or so differently if the x is a minus 2 then this would be a minus minus 2 or a
positive 2. and the same thing if we go down here we
have symmetry with respect to the x axis so this would be the point right here
it would be the same x and this right here would be negative y
and then we have the same point over here symmetric with respect to the x and y
axis with respect to the origin so this would be the point here negative
x negative y so the point is is that if you know a point on the unit circle

00:04
in the first quadrant then we can get similar points by using
symmetry we can get more points on the unit circle using symmetry
all right um and so what we’re going to do now is you know
finding more points on the unit circle besides just these four right here
so let’s see how to do that how to find points on the circle and first one is um
if x is one half what is the y all right so let’s um
you know draw another circle over here just kind of guess at it so
this right here is 1 right so x could be one half right so if they say if x is 2
then what is y well there is no y because when x is 2 it’s not just simply
not going to be on the unit circle right
and so you know i’m asking this question if x is y what is y on the unit circle

00:05
right so what would the y be up here unit circle x squared plus y squared
equals 1. so the point is is that if you if you
know one of the x or y you can find the other so if x is one half
then for this part right here a we’ll have one half squared so that’s the x
squared and then plus y squared equals one squared
and so we can solve this so this would be one minus what is this one fourth
and then you know just subtracting think of the one is four over four so we get
four minus one so we get three over four and now we can take square root of both
sides we’ll have plus or minus square root of 3
and then square root of the 4 we’ll have a 2 here
so it’s important to keep in mind that we’re going to get two y values in other
words when this x is one half here and i’ll kind of line it up a little bit
more better when x is one half here we’re going to get this uh y right here

00:06
this y value right here and we’re also going to get this y value right here
and so this will be the square root of 3 over 2
and this will be the minus square root of 3 over 2 right here
and yeah and so we’re going to get two output values for this x we’re going to
get two y values right here so you know we can ask we can switch the
question what if y is now one half so what would the x be
and this is kind of fun about the unit circle is there’s so much symmetry here
so now i’m going to start off with x square plus and then now y is one half
and so this is basically solving the same equation we just did a
second ago but the only difference is now we’re we have an x missing right so
we have x squared equals one minus a fourth x squared equals three fourths
so x equals plus or minus square root of three over two by the way um

00:07
it’s not a bad idea when when when you’re working in trigonometry to have rest f
rough estimations of these numbers right here right
because everything is taking place between minus one and one
and minus one and one and so you know it’d be kind of nice to
know an approximation for this i think it’s something like 0.8
6 or 7 or something like that so it’s a good idea to have a rough estimation
of these numbers here but you always want to work with the
equal sign with the exact value all right anyways um what about if x is
here’s another one here um this one is about what 0.707 something like that
so this is like um by the way here um just a little side note here
you know because some people write it like this
but some people will write it like this why are these two things actually

00:08
the same so let’s just look at this right here if you go to a calculator you get
approximately 707 something like that um but let’s
rationalize it square root of 2 over square root of 2
and then in the numerator we get square root of 2
and then the denominator we get right and so this is the same thing as
multiplying by one in other words in other words these two things are
equal to each other now you might ask um why rationalize why do that
well there’s a lot of interesting um things that can be accomplished using
rationalization whether you’re rationalizing the denominator
or if you’re going to rationalize the numerator you know i could rationalize
the numerator here we could reverse this process simply multiply by 1 again but
make it square to 2 over square root of 2 and then this will be 2 over 2 square

00:09
roots of 2 but you see the 2’s cancel so if you want to start with this and
rationalize the denominator now you get the denominators no more radical and you
can you know rationalize the the numerator also in any case the point is is that
these two things here are equal and there are a lot of different advantages for
rationalizing especially when you’re working with algebra and not just real
numbers like this but you know there’s some advantage to this
so it used to be before people used computing devices so much before they
were readily available so screw to 2 could be in the back of a
book in a table and you might want to look it up or something and you’d have
something like 1.4 zero [Music] you know some some approximation to that

00:10
maybe it’s another one here but the point is is that if i could go look this up
now i can chop that in half right that’s that’s about 0.7 right so knowing
this numerator right here in a lookup table
then i can cut it in half or a third or you know i can do something like that
and i can get an estimate for this just knowing an estimate for this
whereas it’s much harder to even if you’re given a rough estimate for scrota 2
it would be you know not too hard but in any case rationalizing it’s important
um and so i’m going to start off here with x square root of 2 over 2 what is
the y so this is you know roughly 0.7 in other words it’s between
-1 and 1. so this question makes sense there is going to be a y
so let’s do that so we have x squared so x squared
plus y squared equals 1 squared so y squared equals 1 minus

00:11
sorry and now let’s move let’s square this and move it over
so this top here will get a square so that’ll be two over four
or one half if you will i’ll just say you know two over four
and so this will be think of this one here is four over four so this will be
two over four or said differently one half so that’ll be two over four
one half one minus a half is a half and so now we’re getting this um
yeah so we’re going to get 1 over square root of 2.
of course we need plus or minus and now if we rationalize we’re going to
get square root of 2 over 2. so however you want to think about that
so that would be the y’s for this x so that’s interesting when the x is
square root of 2 over 2 the y turns out to be plus or minus

00:12
square root of two over two similarly if we have y equals square
root of two over two what will the x be so now if we put the
y as square root of two over two and ask what is the x
right so then now this will be the x right and so this will just all switch so
speeding up the process here x squared plus and then the y squared so y
squared equals 1. so this will be x squared equals 1 minus and then now this
will be one half and so we get one half and taking square root
we get 1 over square root of 2 which is square root of 2 over 2. so
uh that that’s nice those uh are all nice and now you may ask the question
um is there a pattern to all this so that’s the question i’m asking is there
a pattern to all this and so let’s try to make the pattern here um

00:13
so let’s see here um think about it like this think about it
as square root of zero over two square root of 1 over 2 and this is for the x’s
and this is for the y’s and we have square root of 2 over 2
and we have square root of 3 over 2 and we have square root of 4 over 2. and the
y’s are going to be reversed so this will be square root of 4 over 2
square root of 3 over 2 square root of 2 over 2
square root of 1 over 2 and square root of 0 over 2. so
you know obviously these simplify square root of 0 over 0 that’s just 0.
this is the pattern though you can see the pattern the numerator is just going
from 1 to 4 and on the y it’s the opposite it’s going back down backwards
right but these reduce obviously square root of zero over two that’s just zero
and square root of one that’s just one one half
square root of two over two square three

00:14
over two and then so this will be square so this would be 2 over 2 so this will
be 1. so the x can go from 0 to 1 and then the y’s will reverse 1 square 3
over 2 cannot reduce not reduce this is one half and this is 0.
and so yeah there’s the pattern right there and we’ll see this uh happening here
when we start uh making drawing out the inner circle so
um what are special angles now so that’s the next question here is what
are special angles because we’re doing trigonometry now in the previous episode
we talked about uh you know trigonometric angles and the difference between
a trigonometric angle and a geometric angle
and we talked about how trigonometric angles measure rotation
and we talked about converting between radius and radians in degrees so i’ll

00:15
assume you’ve watched that video link is of course below in the description
but you know converting between radians and degrees and having lots of practice
with that will be indispensable um you know
let’s talk about the special angles now what makes them special and what are
they so here we go so we’re going to look at multiples of 30 and multiples of 45
and so we’re going to start off with 0 30 degrees 60 degrees 90 degrees we’re
going to keep going until we get to 360.
and those are going to be called special angles
and we’re going to go also multiples of 45 degrees so 0 degrees 45 degrees 90
you know all the way around the circle until we get back to 365 degrees
uh now that’s stated in terms of degrees but we’re also going to think about it

00:16
in terms of radians so if you convert 30 degrees to radians you get pi over six
so we’re going to be looking at multiples of pi over six one pi over six
two pi over six three pi over six four pi over six
now of course there’s some overlap here when you’re looking at multiples of 30
degrees you’re going to hit 90 degrees and when you’re looking at 45 degrees
multiples you’re also going to hit 90 degrees so there’s some overlap
and yeah and so if you convert 45 degrees to radians you get pi over four
and so we’re going to be looking at multiples of pi over four
and so these are going to be called a special angle and you know the question
is why not others well that’s a good question
if you start looking at multiples of say every five degrees
then that’s just going to be really too much information
to memorize and that we don’t really need we don’t really need all that we
can find um you know we can work with other smaller degrees or larger degrees uh
larger angles uh just knowing these basic few so

00:17
it’s sort of it’s sort of like um you know when you’re thinking about
multiplication for example seven times eight
how do you do seven times eight well there’s two ways to do this the first
way is well you just have it memorized right because you memorized
multiplication tables up to a certain number for example maybe 12 by 12
you know 1 through 12 you have the multiplication tables and so 7 is less
than 12 or or equal and 8 is less than or equal to 12 and so this is just a
fact that you have memorized but you know the meaning of it
this is just eight plus eight plus eight plus eight seven times right so there’s
four five six seven times so i’m just going to add up these two and get 16 and
get another 16 get another 16 and then add an eight and then add these
two together i mean you’re just going to get 56 right here right one two three
four five six seven of them and you’re going to add them up and then

00:18
you’re just going to get to 56 well you don’t want to have to do all this every
time you want to have a small number of them memorized
now we’ll have a process for multiplying larger numbers right like how do we do
this right so knowing 3 times 2 is 6 you know
so knowing the basic multiplication tables memorized now we can extend it
we’re going to do the same thing with angles we’re going to have something
called special angles multiples of 30 and 45 degrees
and knowing those special angles knowing the points on the unit circle
corresponding to those special angles then we’ll be able to generate a
procedure that we can do a lot more with all right and so it’s going to be very
incredibly important to memorize the special angles what they are and what
the points on a unit circle correspond to so let’s go ahead and construct a unit
circle um you know and we want the goal is to do

00:19
this uh in less than five minutes so i’m going to show you how to do it the
first time the first time you do it it’s obviously gonna take more than five
minutes but the goal is to be able to do it in less than five minutes so for
example when you’re multiplying uh trigonometric uh sorry when you’re
multiplying uh when you’re learning your multiplication tables you know you’re
not the fastest in the world at it you know you’re a young person you’re
learning it but eventually you want to get this down to just a split second
right what is six times nine you know you just need a split second you don’t
need to go think about it so you want to
have the same thing with the unit circle with the special angles so the first
time we construct it will take us more than five minutes but you want to keep
practicing this over and over again until something that you can do in less
than five minutes so the first thing i’m going to do is use this right here to
draw me a decent circle so let’s go around here and get a circle
and let’s remove this now all right and there’s a somewhat decent

00:20
circle and i’ll try to eyeball the center here let’s say it’s about right here
and now let me draw my coordinate axes and i’m going to use a straight edge
now when you’re doing this by yourself and you’re just you know practicing you
don’t need to draw a perfect circle you don’t need to draw perfectly straight
lines but you know i wanted to show you this in a reasonably nice way so i use
something to help me draw the circle and draw the lines but ultimately when
you’re practicing you just make a circle in fact practice making a circle that’s
actually you know something to to be you know have fun with but anyways let’s
do multiples of 30 degrees so i’m going to start here by drawing uh 45 degrees
first and so i’m going to try to chop this in half as best i can
let’s go about right there and let’s try to chop this in half here too

00:21
this is going to give us our multiples of 45 here and then we want to chop this
so i want to go to about 30 degrees right about here
this will give us about our 30 degrees here
and so this will be 30 degrees and then we’ll come up here and do about 60
degrees here so i know this isn’t perfect i’m not
measuring the exact angles but that’s that’s going to be close enough right
there now this is the unit circle so the equation is the um
let’s put it over here in fact let me move over here let me move down here and
let’s put the equation right here is x squared plus y squared equals one that
tells everyone that we’re looking at the unit circle here
now what are these points we practice finding these points on the unit circle
oh i forgot the multiples of all the 30 degrees right so
we need to come through here and get this quadrant here also let’s go through

00:22
here about like this and then the last one right here all right
so now this is the point right here um one zero so i’ll just call this one zero
here in fact let me put the points here in a different color
just to make it perhaps a little bit easier to read
so this will be the point here one zero this will be the point here
square root of three over two one-half and this will be the point here square
root of two over two square root of two over two
and this is the point here one half square root of three over two
and one zero oh sorry zero one and now you know all the symmetry that we know
we know what this point right here is so
we know what this point right here is we just need to switch the x so i’m going
to say here i’m going to put them up here minus one half

00:23
square root of three over two and then this is the point right here
minus square root of two over two square root of two over two
and this is the point right here corresponding to this one right here
minus square root of three over two one half so that so these correspond to each
other if i measured it out exactly by the degrees here they would look more
symmetric that one doesn’t quite look looks a little bit too high but i think
we get the point across and now this one right here is
corresponding to this one right here the symmetry across the x-axis so this this
is the point right here um the same x square 3 over 2 and then minus one half
and this is the point here square root of 2 over 2
minus square root of 2 over 2. and this is the point here
um one half corresponding to this point right here one half

00:24
square three over two with a minus on it and then so on we can get this last
quadrant right here so this will be the point right here um minus 1 0
and this will be the point right here minus square root of 3 over 2
minus one half and we’re going to get minus square root of 2 over two
minus square root of two over two and then the last point right here we’re
going to get from coming from down from this one right here we’re going to get
the minus one half and minus square root of three over two
okay and then last one right here is we’ll just say that this point right
here is uh 0 minus 1. here we go so there’s all the points on the unit circle
now what about the angles so this will be the angle right here
i’ll put it right here 30 degrees 45 degrees
now i’m going to actually not go around once i’m going to go around twice i’m

00:25
going to do 30 degrees 60 degrees 90 degrees then i’m going to do 120 degrees
i’m just counting up 30s 130 degrees 230 degrees 330 degrees 430 degrees
530 degrees 630 degrees 730 degrees which is uh 210 and then 240 and then 270
and then 300 and then 330 and now i’m going to do the four multiples of 45
so 45 degrees and then 90 degrees we already got that one and
then let’s add another 45 so 135 degrees and then
down here we’re going to add another 45 so we get to 180 and then we’re going to

00:26
add another 45 and get to 225 right here 225
and then add another 45 to get to 270 which we already have labeled and then
right here we’ll get to 315. 270 plus 45 and then last but not least
is you know 315 out of 45 we get to 360. so there’s all of our degrees there’s
all of our points and then lastly we need the radians
so i’ll just say pi over six that’s one pi over six this will be two pi over six
three pi over six this will be four pi over six which
reduces to three pi over four and this will be um so that will be
uh six pi over six um [Music] uh i got confused what are we doing here
we’re doing every 30 degrees so sorry 1 pi over 6 2 pi over 6 3 pi over 6

00:27
4 pi over 6 which is 2 pi over 3 and that’s right here and then 5 pi over
6 right here i’ll put it right here and then six pi over six and then seven
pi over six so seven pi over six i’ll put right here
and then here we get eight pi over six which is four pi over three
and then we’re going to get nine pi over 6 which reduces to 3 pi over 2 and then
so we’re going to add here 9 pi over 6 so 10 pi over 6
which reduces to pi over 3 and then we have 11 pi over 6 right here
and then the last one is 12 pi over 6. all right and now let’s do every 45
degrees but let’s do it in radians so we have pi over 4
so basically we need to count up our pi over fours one pi over four two pi over
four and then this right here is three pi over four

00:28
and then 180 degrees is four pi over four and then we have five pi over four
and then we have six pi over four which reduces to three pi over two and then we
have seven pi over four um and then last we have the eight pi over four
or two pi so there we go and you know it’s really crowded
um it probably would look better if we didn’t um
draw the actual lines there but we we got the actual rays in there to make the
angles here’s 45 degrees they’re 60 degrees and so on we got all that
we got all that in there all right so can we get that down to five minutes or
less and can you get it organized such a way that it looks nice
all right so got to practice that right so the next one is

00:29
the next question is why only go around once why not go around say
you know there’s my red so why not go around say four times
and then stop somewhere why not have that angle on there
well it’s still going to give us the same point
right if i went around four times and then i went around 90 degrees more
right so that would be four times 360 degrees plus 90 degrees right
that’s a large angle well it’s not too large but you know
that would give us the same point right here
um but the angle would be a lot bigger but in terms of the unit circle it would
give us the same point on the unit circle that the terminal side would still be
the same the initial side is still the same so in some respects these two angles
this big angle and this 90 degrees here give us the same point on the unit
circle and so when we’re looking at circles and triangles

00:30
um the points on the triangle on the circle are going to matter
so the the trig not the trigonometric angle is important because it measures
rotation so that’s important what we’re going to see is that if we go around the
circle one time only and we get all this
information down then we’ll learn how to use this information all this
information that you see here and work with larger angles
and the key idea there will be something called reference angles
so what are reference angles so let’s get this circle on here now and
let’s get rid of this here real quick so reference angles are going to allow
us to take a a small group of memorized angles or special angles
and extend what we know about them um so this is you know 30 degrees right

00:31
here i’m not drawing my rays anymore it’s 30 degrees and this is pi over 6
and this says square root of 3 over 2. now um i’m wondering if i can
make this maybe a little bit larger for us real quick
so let’s see here this is going to be um uh y only go around once and
let’s change this to say something a little bit bigger
i’m not sure if you can see these angles
there that’s there’s a little bit bigger um and then try to bring the
label back here down here uh okay so wrong way there anyways um yeah so
now i can bring this to like a bring this up a little bit more so we

00:32
can see the label there all right so that looks a lot better all right so
you know this right here is 150 degrees and we get minus square three over two
and one half for example um and so we got this um angle coming
right through here and here’s the angle right here
and i’m going to put the reference angle in red so the question is
what are the reference angles so this is the reference angle right here now
when i go around let’s say i’ll draw another angle in purple here let’s go
around once and then i’ll go around again so that will be 360 plus 150 so zero

00:33
you know one four five five hundred and ten
that’ll be 510 this angle right here is 510 degrees
so it has the reference angle of 30 degrees also
so this is 30 degrees here right because it’s 150
and then we need 30 more to get to the to the horizontal axis this is 30
degrees here so for 150 degrees and 510 degrees they
both have the same reference angle and we’re going to talk about reference
angles now and the idea the important uh idea of the reference
angle is it allows us to extend our base of knowledge the part that we’re
memorizing and extend it out to allow us to find trigonometric angles
much quicker for larger angles so later on we’ll get to what are the trig
functions we haven’t talked about trig functions yet in this series but later

00:34
on you might have seen them before later on we’ll talk about the sine of 510
degrees and we’ll be able to break that down
by looking at a sign goes there will be a sign here
which i’ll talk about i’ll show you how to find that later
and it’ll be sine of the reference angle so there’ll either be a plus or minus
here so the point is is that we’ll be able to
have our knowledge of special angles and
our trig functions of special angles and then find trig functions of
larger or smaller angles so that’s the basic idea behind reference angles
so uh let’s take a look at what what it is actually what are reference angles so
we’re going to say we have a trigonometric angle in standard position
right so we talked about trigonometric angles in the first episode but
basically where’s um the angle the vertex of the angle is right here at the
origin and the initial side is along the horizontal axis

00:35
and so then we’re going to say the reference angle is the acute angle
between the terminal side and the horizontal axis so for 150 degrees the
reference angle was right here was 30 degrees and the reason why is because
this is the terminal side right here that’s the terminal side of the angle
that’s the initial side and this is the terminal side right here
so terminal side is right here and so the reference angle this is 150
the reference angle of 150 degrees is the acute angle between the terminal side
and the horizontal axis so what’s another example like let’s go down here to
say 225 right here if i can draw that out reasonably well here
let’s go right through the origin and then 225 right there

00:36
there it is right there so let’s just go right through there
now what’s the reference angle this angle right here is 225 degrees what’s
the reference angle so it’s the acute angle i’ll draw in red
it’s the acute angle um between the x-axis right here or the horizontal axis
whatever you want to label it and the uh terminal side right there
so horizontal axis to terminal side and it’s so the it’s always going to be
an acute angle so it’s always between it’s always positive between you know
between 0 and 90. um and so you know this is just 45 degrees right here i’ll
label that in in red right here that’s 45 degrees right there so 225 has
reference angle 45 degrees so i like to abbreviate our a for reference angle of
225 degrees is 45 degrees right that’s 45 degrees right there

00:37
it’s 180 and then plus 45 degrees more gives us the 225 right
so we can also do uh negative angles so let’s look at
another one just real quick so what if we go down from here and
let’s go to here let’s see if i can sketch that real quick uh there we go
and now what will be this reference angle right here
so the angle that i’m looking at here is negative now so negative what
so this will be 60 degrees right here so this is negative 60 degrees
or if you wanted to say positive 315. so we can also do that one at the same
time we’ll do two angles that right there is 315. so that’s 315 degrees
so the reference angle so what is this angle right here this acute angle right

00:38
here this acute angle right about here it’s positive 60 degrees right that’s
positive 60 degrees so the reference angle of 315 is
60 degrees and the reference angle of minus 60 degrees is 60 degrees
all right so that’s the reference angle there
terminal side the horizontal axis it’s always an acute angle and you know you
can find the reference angle of any angle that’s not quadrantal if it as
long as it doesn’t lie right upon an axis all right so um let’s look at a couple
more examples um let’s look at some more examples here let’s look at

00:39
30 degrees first and let me move up here so what’s the
reference angle for 30 degrees now this is easy but i just want to point it out
and make sure that everyone’s on the same page so here’s 30 degrees right here
and reference angle is just 30 degrees so that’s it reference angle of 30
degrees is 30 degrees in fact for any acute angle between 0 and 90
it’s its own reference angle right so all the special angles that we know
of between 0 and 90 in the first quadrant 30 45 60
they’re their own reference angles all right what about 135 degrees
so let’s draw it right here here’s 135 degrees there it is
and the reference angle is right here it’s 45 degrees

00:40
so let’s just write that down reference angle of 315 or sorry 135 is 45 degrees
that’s it all right let’s look at another one about 240. so let’s draw 240
let’s put it right here where’s 240 so we’re going to go to 180
and then how much more do we go we’re not going to go to 270 so we’re going to
go to 240 down about here let’s just say about there there’s 240 degrees
and so what’s the reference angle so the reference angle is right here
and so i need to take away 180 because we’re not going to count this 180 as
part of the reference angle so to do this i just simply need to do uh 240
minus 180 and that gives us 60 degrees right here so the reference angle is 60

00:41
degrees we’ll just put that here reference angle of 240 is 60 degrees
all right good let’s do the next one 330 degrees where’s 330 degrees
it’s in quadrant 4 right so i need to go to 270 and then a little bit more
and so there’s my 330 degrees and is this the reference angle right
here no it’s always the acute angle with the x-axis so or horizontal axis um
so yeah so what is the what is the missing right what is it missing
so now to say what’s missing is i’m going to say it’s 360 and then take away
that right there right and that’ll give us 330 right so this is going to be 30
degrees right here so the reference angle reference angle of 330 degrees

00:42
is 30 degrees all right so far so good minus 210 degrees
so let’s do that one about right here where’s minus 210 so now i’m not going
in the positive direction i’m going in the negative direction i’m going to go
negative 180 and then i’m going to go a little bit more i’m not going to go
negative 270 so i’m going to stop right here and that’s going to be negative 210
degrees now like i said i’m not going to go negative 270 so i got negative 210
and then going straight to the y-axis would have been negative 270. so what’s
left over here is the difference between 270 and oops sorry
that’s the wrong axis there it’s this one right here there’s the reference
angle right there so i went to 210 i went negative 180 and then i had to go
more about 30 degrees so this is going to be 30 degrees right here

00:43
so here we go reference angle of negative 210 is 30 degrees all right great
next one minus 140 let’s do that right about here
where’s minus 140 it’s not quite to minus 180 is it so there’s -140
and what’s left over right here is 40 degrees so 40 degrees right here if i
did minus 140 degrees if i had gone another minus 40 i would have been minus 180
so that’s why that’s my that’s why that’s 40 right there
right so the reference angle of minus 140 degrees is 40 degrees right there
all right so let’s do another one let’s do a one in radians now

00:44
so five pi over three so quick do you remember where five pi
over three is right so this was a special angle that was on our unit
circle and so it’s very important that you know where that is right away as
fast as possible because you know you’re going to be doing a lot of trigonometry
and to have to slow down and think and count
up your pies to figure it out right so here’s how i would figure it out if i
had to start from scratch and i don’t remember it here’s one
sorry here’s 1 pi over 3 well here’s pi over 6 here’s 2 pi over 6 3 pi over 6
4 pi over 6 5 pi over 6 6 pi over 6 7 pi over 6 8 pi over 6 9 pi over 6
and 10 pi over 6 which happens to reduce to 5 pi over 3.

00:45
so this is 5 pi over 3 right here and as you can see is i went to 270
and then i went 30 more so that’s 300 degrees right there if we
were to convert 5 pi over 3 into radians we would get 300 degrees right there
but in any case we’re doing this in radians so let’s just draw the angle
put it right down about right there and so we got 5 pi over 3 is the angle
right there i’ll label it up here 5 pi over 3 and now what is left over
so to find what’s left over we can do 2 pi that would be all the way around
take away what we already did and then this will be the remainder right there
so this will be 2 pi all the way around and then take away what we already did
and so then we get left the reference angle now think of 2 pi is 6 pi over 3

00:46
and so we get pi over three and so that’s the reference angle
so i’ll just put r a right here reference angle and um the last one here
is 870 degrees i guess we can try to put that one right here where’s 870 degrees
so let’s go around here many times well not many times how many
times we’re going to go around we’re going to go around once that’s 360.
we need to keep going though go around again that’s 720 so what is 870 minus 720
that’s 150 degrees right there right so we need to go 150 degrees more
so it went around 720 two times and then 150 degrees more
so this will be 870 degrees and i’m going to end up right there
and then we can see that the reference angle right here is 30 degrees

00:47
and the reason why is because yeah we went around twice but then we
went around 150 degrees more so 150 degrees more to get to 180 the straight
the horizontal axis that we would have had to have gone 30 degrees more so
what’s left over there is 30 degrees so the reference angle so reference angle
here of 870 is 30 degrees there we go um and so
if you enjoyed this video i hope you like and subscribe and
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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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