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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 were an angle terminates,
and specifically I want to deal with angles that are measured in radians.
So let's get started. Suppose you get a problem like this:
Determine the quadrant in which each of the following
angles terminates: 3pi over 4,
and then 4 radians,
9pi over 4, 7pi over 2, and 7 radians.
So I'll draw a coordinate plane.
Here's the x-axis and y-axis,
and I'll mark the four quadrants.
And
if we have an angle beginning in the standard position, in other words,
starting
on the positive side of the x-axis,
that's going to travel, while it's in the first quadrant,
from zero radians up
to pi over 2 radians.
After that, it's going to go through the second quadrant,
until it's covered pi, or pi radians.
It then goes through the third quadrant, which would be
covering 3 pi over 2.
And then if it makes a complete circle counterclockwise
around the origin, it's going to end up at 2 pi.
So I'm asked to find out where
an angle that measures 3 pi over 4
is going to terminate. So it starts out in standard position,
it travels through the first quadrant,
that's 1/2 pi. I wanna go 3/4 pi.
Well, 3/4
is between 1/2 and one, so that means I must end up
somewhere in the second quadrant.
In the next problem I'm just given 4,
in other words, 4 radians. So what I have to do here is remember
that pi equals approximately
3.14. So if we go through the first
and the second quadrants, that will take us
3.14 radians. I want 4, so I'm going to end up
in the third quadrant. For the next one I've got
9 pi over 4.
Well, if I went
all the way around through the first, second, third, and fourth quadrants and ended up
at 2 pi,
2 pi would be the same
8 pi over 4.
8/4 is the same as 2.
So that would mean I had gone
8 pi over 4, and then continued...
let's see... 9 pi over 4
8 pi over 4
would be just 1 pi over 4, 1/4 pi.
So I'm going to end up in the first quadrant.
In other words, I've traveled a complete circle -- that's 2 pi,
or 8 pi over 4. I want to find out how much further I've gone,
so I subtract that 8 pi over 4 from 9 pi over 4,
and I end up here in the first quadrant.
Let's do a few more.
So what about 7 pi over 2.
I looks like, once again, I'll go a complete circle around.
If I travel all the way around and cover to 2 pi,
2 pi is same as
4 pi over 2. In other words, 4/2 would reduce down to 2.
So if I subtract that 4 pi over 2
from 7 pi over 2,
that leaves me with 3 pi over 2.
Let's see...
that takes me down...
I've got a full circle around, then I go through the first quadrant,
second quadrant, and at the border between
the third and fourth quadrants, I get to 3 pi over 2.
So an angle that measured 7 pi over 2
would take me to that border between the third and the fourth quadrants.
And for the last one, once again I've got
a radian measure, 7.
So let's try that. (I'll clean this up a little bit.)
So remember once again
that pi equals approximately
3.14. If I want a complete circle around,
that would be 2 pi.
So 2 pi is going to equal approximately
6.28. But I want 7.
So I'll go a little further than 6.8.
That means I'm going to end up once again in the first quadrant.
So that's the basic idea. I hope this helps.
Take care. I'll see you next time.