WEBVTT
00:00:00.880 --> 00:00:08.660
Determine the quadrant in which π lies if cos π is less than zero and sin π is greater than zero.
00:00:09.670 --> 00:00:11.650
Letβs have a look at the unit circle.
00:00:12.960 --> 00:00:17.390
There are four quadrants of the graph just above the positive π₯-axis.
00:00:17.390 --> 00:00:26.030
There is the first quadrant, and then moving counterclockwise, we get to the second then third then fourth quadrants before weβre back to where we started.
00:00:26.870 --> 00:00:30.510
We take an arbitrary point on the unit circle in the first quadrant.
00:00:31.610 --> 00:00:36.670
The corresponding angle π is measured counterclockwise from the positive π₯-axis.
00:00:38.080 --> 00:00:44.150
And the nice thing about the unit circle is that the points on it have coordinates cos π, sin π.
00:00:45.100 --> 00:00:56.190
And in considering the coordinates of this point, we can see that both cos π and sin π are greater than zero, and in fact, they will be for any point on the unit circle in the first quadrant.
00:00:57.500 --> 00:01:13.520
We can do the same for another arbitrary point, this time in the second quadrant, and we can see that the value of cos π is less than zero β itβs on the negative π₯-axis β but the value of sin π is still greater than zero β itβs still on the positive π¦-axis.
00:01:14.440 --> 00:01:23.760
And it appears that this is the quadrant weβre looking for; weβre looking for the quadrant in which π lies if cos π is less than zero and sin π is greater than zero.
00:01:24.410 --> 00:01:27.770
And we found that these two things are certainly true if weβre in the second quadrant.
00:01:28.450 --> 00:01:33.520
We can check that, in the third quadrant, cos π is less than zero, as is sin π.
00:01:34.730 --> 00:01:40.620
And in the fourth quadrant, cos π is greater than zero, but sin π is less than zero.
00:01:41.550 --> 00:01:58.350
So going back to our question where we were asked to determine the quadrant in which π lies if cos π is less than zero and sin π is greater than zero, we can see that this only happens in the second quadrant, and so thatβs our answer: π lies in the second quadrant.