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.