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>> This is YourMathGal,
Julie Harland.
Please visit my website
at yourmathgal.com,
where all of my videos are
organized by topic.
This is part two
of degree measures of angles
and we're going to be talking
about positive
and negative angles,
quadrantal angles,
and some commonly-used angles.
So, the degree measure
of an angle is positive
if its rotation is counter
clockwise and vice versa.
So, if you have a positive
angle like 30 degrees,
you have a counter clockwise
arrow here.
Okay, and if you have a
counter clockwise arrow then
we're talking
about a positive angle.
The degree measure
of an angle is negative
if it's rotation,
this little arrow here,
is in the clockwise direction
and vice versa.
So if you see negative 45
degrees, you've got
to make sure you're going
in this clockwise direction,
okay?
So that's all there is
to whether it's positive
or negative.
Okay, given now--
let's try an angle
of 120 degrees
and negative 90 degrees.
All right,
let's do 120 degrees first.
So what I think
about this is 120 is 90 plus
30 more, so if I think of my X
and Y axis set up here,
just easy as for me to do,
I think, well
if I go 90 degrees
and I've got up to here,
but I got to go 30
more degrees.
So, that means--
so here I'm starting
and how would I end
about right here.
Okay now, I'm just sketching
these and I say draw--
it's not exact, we have to get
out for a factor but--
it's in the positive direction
so I went up here,
120 degrees.
Now if you--
you could also just free draw
something like this, okay?
I didn't say you had to do it
on the X and Y axis.
That just tells me to kind
of measure
that 90 degrees plus 30
degrees mark.
Right. How
about negative 90 degrees?
All right.
So, again let's go ahead
and think about what it looks
like for looking at our X
and Y axis and let's say,
we're starting here, right?
We have our initials right
here, and then we are going
in the negative direction,
so we're going down, right?
We're going to go
down 90 degrees.
That's of course, either way,
and that would be negative
90 degrees.
So, remember
when it's a 90-degree angle,
you could put--
you could square it off
with little square, okay?
Another way to try it is just
over on this side,
you're showing
that it's negative 90
by drawing in that arrow,
that's how you could tell.
Now, if an angle is
in standard position
like we did
on the previous page,
where we put the initial side
right on the X axis,
the right side
of the X axis, right?
We say it, the angle,
lies in the quadrant
where its terminal side lies,
all right.
So what am I saying?
All we do is we look
at where the terminal side
of an angle lies and we say,
"Oh, that angle is
in that quadrant.
"So let's look back.
So if I look
at 120 degrees here,
in this position, right?
Where does its terminal
side end?
So, for 120 degrees,
we see the terminal side
of 120 degrees ends
in quadrant.
Okay, now let's see.
Do you remember
where the quadrants are?
Remember, if you've got your
quadrants, that you have
to know as one, two, three,
and four, and here it is
up here in quadrant two.
So the terminal side
of 120 degrees, you know,
that angle,
ends in quadrant two,
I'm sorry, in-- yes--
in quadrant two.
So, instead of saying,
all that we just say 120
degrees is
in quadrant two, okay?
Or lies in quadrant two, so,
we say 120 degrees lies
in quadrant two.
So, you're really looking
at the terminal side.
But we don't write all
that out, okay?
Same thing, if you want
to figure it
out for negative 90 degrees,
all right?
So the terminal side in 90--
negative 90 degrees, well, oh,
it's not in a quadrant,
it's right here on the Y axis.
So the terminal side of 90 is
on the Y axis, right?
So that means negative 90 is a
quadrantal angle.
[ Pause ]
I'm not sure I have said
that yet, but here it is,
I think I quit, I had a time.
If its terminal side is
on the X or Y axis,
it's called a
quadrantal angle.
So 270 degrees is a quadrantal
angle as shown because,
just like negative 90 degrees,
its terminal side, right?
There is this terminal side
right there.
The terminal side is right
on the Y axis.
This would also be true
of 180 degree angle 'cause its
terminal side is right
on the X axis.
All right.
Now, when you're learning
trig, there's an angle
that come up over
and over again
and those are multiples
of 30 degrees and 45 degrees.
And so I've put some angles--
all those angles
that are multiples
of 30 degrees and 45 degrees
that happen to be
in between zero
and 360 degrees.
So, I've only got positive
angles here.
And so, you could see
that 30 degree angles here,
like every 30 degrees, 30, 60,
90, 120, 150, 180, 210, 240,
et cetera.
And then the 45-degree angle,
that's halfway in between 90,
so those are the ones right
through center,
I did it with the green color,
45, 90, 135, 180,
225, 270, 315.
And if I want one more,
it would be at 180.
Notice I have this negative 30
degrees here.
I just want
to have you notice something.
If I did draw a negative
angle-- negative 30 degrees,
it ends up being
in the same place
where 330 degrees is.
So, you could see,
I could've drawn a lot more
in here, right?
It goes on forever.
So, I will start repeating
myself at some point.
So, and what it did for you
to be able
to reproduce this figure
on your own, so you might want
to try that.
And notice like, you know,
this is just a sketch,
these aren't exact.
My lines don't even look
like real lines.
I'm free-hand drawing it.
So this is 30 degrees
and it's 15 degrees here
and 15 degrees here, right?
From 30 to 45
and from 45 to 60.
So, these are kind
of close together
and you got 30 degrees again
and you got between 90
and 120 that's 30 and then 15
and 15 and then 30 again.
So there's kind
of a space here, okay?
And when you look at this,
you can see that 30 degrees,
45 degrees, and 60 degrees,
they all lie in quadrant one,
Whereas, 120 degrees,
135 degrees, and 150 degrees,
all lie in quadrant two,
et cetera.
And the quadrantal angles you
could see here are zero
degrees, 90 degrees,
180 degrees, and 270 degrees.
Okay, so this is what we're
going to be working with a lot
and we'll be doing negative
angles as well, so multiples
of these angles, you know,
because I only treated the
ones in between zero and 360
like you could have 30 times,
you know, let's say 20,
that would be 600 degrees.
That's going to be somewhere
in here as well, right,
if I drew all of the multiples
of 30, I just stopped
at 330 degrees.
Let's go back
to the beginning.
So, going back one page
at a time here.
We decided
that 270 degrees was a
quadrantal angle.
When we did this,
we wrote that 120 degrees lies
in quadrant two,
and that 90 degrees is a
quadrantal angle.
Okay, so let me just sort
of highlight that.
We came up with
this information.
And if I go back once more,
let's just also go ahead
and do this.
Where does 30 degrees lie?
What quadrant?
Both? So over here
in quadrant one.
And what about this negative
45 degrees, where does it lie?
[ Pause ]
That's quadrant four.
Okay? So that's all there is
to it.
This is YourMathGal,
Julie Harland.
Please visit my website
at yourmathgal.com,
where all of my videos are
organized by topic.