Determine the quadrant in which 𝜃 lies if cos 𝜃 is less than zero and sin 𝜃 is less than zero.
Well, we can see that if both cos 𝜃 and sin 𝜃 are less than zero, then it means that both cos 𝜃 and sin 𝜃 are negative. So, how does this help us? Well, it helps us when we take a look at something called the CAST diagram. The CAST diagram helps us to determine whether sin, cos, and tan 𝜃 are gonna be negative or positive. We can also use it to help us find values as well if we do know what sin, cos, or tan 𝜃 are equal to.
So, when we’re dealing with a CAST diagram, it splits into four quadrants. So, we got first, second, third, and fourth quadrants. So, what it tells us is that in the first quadrant, we have A. What that means is that all of cosine, sine, and tangent of the angles between zero and 90 are all positive. Then, if we move in to the second quadrant, what the S tells us is that only the sine ratio, so sin 𝜃, is going to be positive. So, that means that the sine of any of the angles in this quadrant is gonna give us a positive value. Whereas if we had the tangent or cosine of any of these angles, we get a negative value.
Next, we have the third quadrant where the same is true but this time in tan, so the tan of or tangent of 𝜃. So, the angles in this quadrant are gonna be positive. However, all the others will be negative, so sine and cosine. And then finally, in the fourth quadrant, it’s cosine that’s positive. Okay, so we’ve now got a CAST diagram. The way that some people remember it, starting from the first quadrant again to the fourth quadrant, is A-S-T-C, All Silver Tea Cups, or other mnemonics like that.
Well, what we’re interested in in this question, is the quadrant where both cos of 𝜃 and sin 𝜃 are negative. So, this means this can only be in the third quadrant. And, that’s because in the first quadrant, they’re both positive. And in the second quadrant, sine is positive. In the fourth quadrant, cosine is positive. So, it’s only in the third quadrant where they’re both negative.
So now, just to demonstrate that this is the case, I’ve just chosen some values in each of the quadrants. So, we got 45 degrees, which is in our first quadrant, 135 degrees in the second quadrant, 225 degrees in the third quadrant, and 315 degrees in the fourth quadrant. Well, in the first quadrant, the values of cos 𝜃 and sin 𝜃, if 𝜃 was the angle that we’re using, are both gonna be 0.71. So, they’re both positive. If you calculate them for 135 degrees, which is in the second quadrant, we get cos 𝜃 as negative 0.71 and sin 𝜃 as 0.71. Then, for 225 degrees, so the third quadrant, we get both cos 𝜃 and sin 𝜃 as negative 0.71. So, this was as we expected.
So then finally, in the fourth quadrant, so 315 degrees, we get cos 𝜃 is 0.71, so positive. But sin 𝜃 is negative 0.71. So, it’s negative. So therefore, we’ve determined using the CAST diagram and by substituting in values, that the quadrant in which 𝜃 lies, if cos 𝜃 is less than zero and sin 𝜃 was less than zero, is the third quadrant.