- So in this video, I'm doing exact values of sec, cosec, and cot. Now, in the past, you've used standard triangles to find exact values of sine, cosine, and tan. Now, this is really, really simple as long as you know that sec is 1 over cos, that cosec is 1 over sine, and that cot is 1 over tan. Really, really straightforward. 

So a quick example here, cosec 2 pi over 3, I'm trying to find the exact value of that. Now, I should know that cosec theta is the same as 1 over sine theta, which in this case is 1 over sine 2 pi on 3. Now, you should all be able to visualize that on the unit circle. Quick unit circle here. 2 pi on 3, pi on 3, 2 pi on 3, so over here. 

Now, if I want to put that back into quadrant 1, I've got to rewrite it as 1 over sine pi minus pi on 3. Now, sine minus is sine pi minus pi on 3 is the same as sine pi on 3. 

C-A-S-T. It's in the sine quadrant, so it is still positive. Now, 1 over sine pi on 3. I wonder what pi on 3 is, sine pi 3. So there's pi on 3. Sine is opposite over hypotenuse. So it's root 3 on 2. I'll just move over here a little bit. 1 over root 3 on 2. 

Now, 1 over root 3 on 2 is just the reciprocal of the denominator. So it's 2 over root 3. All right, so a little bit of extra work compared to what you're used to. But it's all based around the fact that cosec is the same as 1 over sine. 

Now, just for the sake of completeness here, I should probably say that your textbook might give a different answer just because they might rationalize the denominator. So they might get to here. That answer's correct. It's not neat though. What textbooks often do is rationalize the denominator by multiplying by the root on top and bottom. Root 3 times root 3 is just 3. So you might see a rationalized answer in the back of your book there. I'd say this is correct, and I'd say this is correct, but we'll talk more in class about that one. 

It's worth noting at this point as well that if you already knew what sine pi on 3 was or if you already knew what sine 2 pi on 3 was, you can immediately jump to 2 on root 3. It's always going to be the reciprocal of whatever the sine was. So sine theta always equals 1 over cosec theta. 

So in this particular case, where if you know that sine 2 pi on 3 equals root 3 on 2, you should immediately know that cosec 2 pi on 3 is equal to root 3 on 2. Those are the things that should just jump at you straight away.