This video is provided as supplementary
material for courses taught at Howard Community
College and in this video I want to show how to determine the
quadrant where an angle terminates. So let’s get started. Let’s say we’ve
been given a problem like this: Determine the quadrant in which
each of the following angles terminates, and then we’re
given a series of angles, 103 degrees, 227 degrees, 503 degrees, and so on.
So here’s what we’re going to do. I’ll draw a rough coordinate plane.
I’ll have the x-axis and the y-axis, and if we want to draw
an angle in that plane — let’s assume it’s gonna be in standard
position — the angle is going to start our on the positive side of the x-axis and, as it continues in a
counter-clockwise direction, it will go through the first quadrant
until it’s gone 90 degrees. Then from 90 degrees to 180 degrees it will go through the second quadrant.
If it continues if will go from 180 degrees to 270 degrees in the third quadrant. And if it continues all the way around
in a full circle it will go from 270 degrees to 360
degrees in the fourth quadrant. I’ll put in zero degrees for
where it started out. So if we’re told we
have an angle that’s 103 degrees, we can just look at this
plane and say it’s starting at zero,
it’s going through the first quadrant —
that’s the first 90 degrees, and then it continues on to 103 degrees, which is between 90 degrees and 180
degrees, so this angle, 103 degrees,
will terminate in the second quadrant.
Let’s look at the next one. Here’s 227 degrees. So now be angle goes
through the first quadrant, goes through the second quadrant.
It’s covered 180 degrees. And it ends up somewhere in
the third quadrant, because 227 degrees is between a 180 degrees and 270 degrees. Now remember an angle can be
greater than 360 degrees. So we’ve got one that’s 503 degrees. What’s happened here is the angle has
gone a complete circle around the origin, it’s covered 360 degrees, and then it’s
gone further, because the complete
angle is 503 degrees. So what I’m going to do is subtract 360 degrees, that’s how far it’s gone the first time
around the origin. When I subtract that, I end up with 143 degrees. So that means that after it went 360
degrees, it went a further 143.
That would take us once again through the first quadrant, that covers 90 degrees more,
and then since who want to cover a
143 degrees after that first 360, we’re going to end up
here in the second quadrant. So all I did was
subtract 360 from this angle,
which was greater than 360, to find out how far we went as we
traveled the second time around the circle. Here’s an even bigger angle.
This one is 756 degrees. So it seems like what must
have happened here is we went one time around the circle and then continue again.
If we go twice around the origin, that’s going to be 720
degrees. So I’ll subtract 720 degrees from the 756. That’s going to leave me
with 36 degrees.
Since 36 is between 0 and 90,
that means it’s going to end up in the first quadrant. So if you’re dealing with positive angles,
what you want to do if the angle is greater than 360 degrees, is subtract a multiple of 360 from the angle you’ve been give
so you end up with some positive number between 0 and 360, and then just use that to find out what
quadrant it terminates in. Let me draw this over again and deal with negative angles. So once again, here’s the x-axis and
y-axis, the first, second, third and fourth quadrants.
We’re starting at zero degrees, then 90 degrees, 180 degrees, 270 degrees, and a full circle,
360 degrees. Now we’ve got -35 degrees. Now you might want to just look at
the coordinate plane and realize that we’re going to end up
in the fourth quadrant. If we start in the standard position
along that positive side of the x-axis, and just go down -35 degrees clockwise, we’ll end up in
the fourth quadrant. But as a general rule you might want to
try something like this. Add 360 degrees to the angle you’ve been given. That will
give us a positive angle, which is coterminal with what we’ve
got. So if we have -35 degrees and we add 360 degrees, we’re going to get
positive 325 degrees, which will take us through the first quadrant, then
through the second quadrant, through the third quadrant.
That’s 270 degrees, and we would end up in the fourth
quadrant in exactly the same place we would have
ended up if we’d gone clockwise, in a negative
direction. Here’s one with
a bigger negative angle. This is -412 degrees. So let’s try that same approach again. I’m going to add 360 degrees to this -412. Oh, that’s not going to be enough, because
I want to add some number which is greater than this negative
amount. So I’ll add 720 degrees to this. When I add -412 degrees and positive 720 degrees, I end up with positive 308 degrees. So what that means is
I can find the quadrant in which this angled terminates
by going counterclockwise through the first quadrant, the second quadrant,
third quadrant. That takes me past 270, and then 308 will end up,
will terminate, in the fourth quadrant. So the
general principles are gonna go like this: if you have a positive angle that’s greater than 360 degrees, subtract a multiple of 360 degrees from it so you end up with some positive number
between 0 and 360 degrees. If you have a negative angle, add some
multiple of 360 degrees to it so you end up with a positive angle
which is between 0 and 360 degrees, and then use the
coordinate plane, or you may be able to just
visualize this, to determine what the quadrant is in which the angle terminates.
So that’s it for this one. Take care.I’ll see you next time.