Video Transcript
Determine the quadrant in which 𝜃 lies if cos 𝜃 is less than zero and sin 𝜃 is greater than zero.
Let’s have a look at the unit circle. There are four quadrants of the graph just above the positive 𝑥-axis. There is the first quadrant, and then moving counterclockwise, we get to the second then third then fourth quadrants before we’re back to where we started.
We take an arbitrary point on the unit circle in the first quadrant. The corresponding angle 𝜃 is measured counterclockwise from the positive 𝑥-axis. And the nice thing about the unit circle is that the points on it have coordinates cos 𝜃, sin 𝜃.
And in considering the coordinates of this point, we can see that both cos 𝜃 and sin 𝜃 are greater than zero, and in fact, they will be for any point on the unit circle in the first quadrant. We can do the same for another arbitrary point, this time in the second quadrant, and we can see that the value of cos 𝜃 is less than zero — it’s on the negative 𝑥-axis — but the value of sin 𝜃 is still greater than zero — it’s still on the positive 𝑦-axis.
And it appears that this is the quadrant we’re looking for; we’re looking for the quadrant in which 𝜃 lies if cos 𝜃 is less than zero and sin 𝜃 is greater than zero. And we found that these two things are certainly true if we’re in the second quadrant.
We can check that, in the third quadrant, cos 𝜃 is less than zero, as is sin 𝜃. And in the fourth quadrant, cos 𝜃 is greater than zero, but sin 𝜃 is less than zero. So going back to our question where we were asked to determine the quadrant in which 𝜃 lies if cos 𝜃 is less than zero and sin 𝜃 is greater than zero, we can see that this only happens in the second quadrant, and so that’s our answer: 𝜃 lies in the second quadrant.
