 Hi, I'm Zor. Welcome to Unisor Education. Let me continue talking about how trigonometric functions behave when arguments are certain basic angles, which we are all familiar with, like 30 degrees, 45 degrees, et cetera. I have already covered sine and cosine functions, and it's important before going into tangent, which I'm going to talk about today, to review how sine and cosine are doing in this particular case. And the reason for this is very simple. You have to remember that one of the equivalent definition of the tangent is this. It's a ratio of sine over cosine. Now, I'm not going to examine the unit circle and different angles separately with triangles, et cetera, et cetera in this case. I'll just use this definition and the fact that I have already examined what are sine and cosine for all these basic angles. So, let me just go straight into this. Now, basic angles are 0 pi over 6, which is 30 degrees, pi over 4, which is 45 degrees, pi over 3, which is 60 degrees, and pi over 2, which is 90 degrees. Now, for these cases, what is a sine? Now, a sine of 0 is 0. It's an originate, right, of the angle of 0 regions. This is pi over 6. Originate is opposite angle, so it's 1 half. So, this is square root of 2 over 2 that I do remember, and I do remember this as well, and sine is equal to 1 for a 90-degree angle. Now, what's the cosine? Well, cosine is just an opposite. It's 1 and 0. So, what is, in this case, the tangent? Well, obviously, it's sine over cosine, right? So, it's 0 in this case. This is 1 over square root of 3, which is the same as square root of 3 over 3. If I'm multiplying by square root of 3, I will get the verticals numerator and denominator. I will get this. This is the more traditional record of this. People prefer to have verticals on the top than a numerator. Now, here, it's 1, because the sine and the cosine are the same. So, if you divide one by another, it will be 1, and it should be square root of 3, and tangent is not defined at 90 degrees. Very simple. So, I'm not going to go into all the details about why how these triangles are arranged. It's much better just to use this particular definition, which is equivalent, as we know. Now, how about other properties? You do remember that sine is an odd function, and cosine is even function. That makes tangent odd function. Now, since it's odd, then we can calculate minus 30 degrees, minus 45, minus 6, minus 90, just reverse the sine, because it's an odd function. Now, if you want something else, you do have to remember a couple of formulas for sine and cosine related to adding pi. Sine of x plus pi is equal to minus sine of x, and cosine of x plus pi equals minus cosine of x. Now, why is that? Well, again, back to the unit circle, which I promised not to mention today. Anyway, it's not mentioned relative to tangent, it's mentioned relative to sine and cosine, so I'm excused. If you add pi from the first quadrant, you go to the third, when both coordinate and abscissa are negative, so that's obvious. That makes the tangent, since both are changing the sine, so the tangent is equal. If you add pi to a tangent, it's exactly the same thing. If you remember, actually, pi is a period of function tangent. That's actually just the proof of this. Since pi is a period, you can derive from these angles, and the fact that the tangent is an odd function, you can derive basically all other nice angles, like 135, which is 180 minus 45 for 270, which is 180 plus 90, or something like this. All angles are basically derivable from all these nice rules. These are properties of tangent for basic angles. Very short lecture, nothing really fancy about this, and the most important tool is this definition of tangent. Thank you very much.