 Hi, I'm Zor, welcome to a new certification. Let's talk about cotangent. Spoke about sine, about cosine, tangent, now it's the cotangent's turn. Well, for whatever reason, historically, sine and cosine are considered like more, I don't know, more main or more dominating functions in trigonometry. Tangent is like on the next level, and cotangent is even lower than tangent in its authority, popularity, and talk about whatever. I don't know why, it's just historically. So anyway, I will try to spend as much time for cotangent as I spent for tangent, basically to address all the issues. So let's just go straight in. All right, definition. We have, I would say, three equivalent definitions of cotangent. One is this, ratio of cosine to sine. Since cosine and sine are defined, then ratio is perfectly well defined. At the same time, another definition is absolutely equivalent, one over tangent of phi. Why? Because tangent is sine over cosine. So if we're aversive, that's the case. And the third definition is unit circle. So if you have a unit circle, you have an angle phi. This is y, this is x. So cotangent phi is equal to x over y. Now, why is it equivalent? Well, because x on the unit circle is exactly the cosine of this angle, and y is assigned by definition. So all these three definitions are equivalent. Most likely you will see this one as the definition, this one. And that's what I will probably use most of the cases. But in any case, just bear in mind that all these three definitions are completely equivalent. Okay, next. Next, what strikes the I is denominator, obviously, which means that the function is not defined where the denominator equals to zero. Now, if sine of x equals to zero, then x, if you remember, is zero, pi, two pi, three pi, et cetera, so it's pi times n, where n is any integer number, including zero and negative numbers as well. So these are the points where our sine is equal to zero and the cotangent is undefined. So the domain of this function is basically a union of interval from zero to pi, not including the edges. That's why I have parentheses rather than square brackets. Then after pi up to two pi, after two pi to three pi, et cetera, and to the negative part from minus pi to zero from minus two pi to minus pi, et cetera. So the union of all these intervals is the main of this particular function cotangent. Or graphically, if you have these points, then these points are excluded from the domain and everything else included. Okay, so let's investigate how the function behaves when the argument is changing. And that's where we will find the periodicity and stuff like this. Well, since I can use any of the definitions for cotangent, I will use this one. So this is x and this is y. So start from the phi equals to zero, this is the interval phi, and I will draw the graph. So start from zero. At point zero, our cotangent, which is x divided by y, point zero is here. It has coordinates x equals one, y equals to zero. So the function is not defined. So there is no value of cotangent at zero. But as soon as we move slightly over point zero to the right, our coordinate is very, very small, right? And our abscissa is somewhere around one, which means that the ratio is very, very large. The closer to zero, the larger would be this ratio. And obviously, as we are moving closer and closer to zero, the denominator goes to zero. Numerator is bounded, it's around one. So basically the whole ratio goes to positive infinity. So that's how graph looks here. At some point, actually when the angle is equal pi over four, which is 45 degrees, y and x have exactly the same value because this is a right angle with a 45 degrees angle. So the ratio would be equal to one. And then as we move forward, increasing the angle to the right angle, pi over two, then the function becomes x would be equals to zero, right? When the point is over there, x would be equal to zero, y would be equal to one, so the ratio would be zero. So it goes and somewhere at pi over two, cotangent goes to zero. Then we go even more and the angle is increased to pi. As angle is increasing to pi, look at the signs first of all. If point is somewhere here, originate is positive, obsessa is negative, which means that the ratio is negative. And it goes from zero, when you go here, you have the originate close to zero. So the whole ratio becomes greater and greater in absolute value since the denominator goes to zero, but it remains negative, which means it goes to minus infinity when we are reaching the point pi, which means that this is asymptote. That's how it behaves. Jump over x is equal to pi to the right. Now, in this area, in this quadrant, both obsessa and the originate are negative, which means the ratio is positive. And around this point, it's an infinite and absolute value because the denominator is close to zero. As it comes to this point, it's the other way around. My originate is around minus one, but my obsessa is around zero. So what's interesting is that in this area, the absolute value goes from infinity to zero, but the sign is positive, which means that to the right of this, we have this closer to pi, closer to pi. It goes to infinity, but positive infinity because both originate and obsessa are negative. And closer to three pi over two, we go to zero and function is defined in this particular point. Next, let's close the loop. We go again to a negative territory of the ratio because obsessa is positive and originate is negative. And since my originate goes to zero, my ratio goes to infinity again. So this is two pi. This is three pi over two. So the function goes like this. So that's how the function behaves from zero to two pi. It does not have any value at zero and pi at two pi. These are asymptotes, and it's zero and pi over two and three pi over two. Now, obviously after that function repeats its value because both sine and cosine or obsessa and originate are repeating their value as soon as we're going to the next loop and the next loop, or if we go to a negative territory, this would be exactly the same thing. If this is minus pi and minus pi over two, it behaves exactly the same. So this is the general behavior of this particular function, cotangent. Defined everywhere except pi times any integer, zero at pi over two plus pi over plus two pi, pi times any integer. And these are asymptotes. Obviously the range is from minus infinity to plus infinity. Okay, what's next about this? What's interesting is although sine and cosine have a period of two pi, their ratio cotangent, which is cosine over sine. It looks like it has a period of pi, not two pi. I mean, obviously two pi is also a period, but there is a smaller period, which is very interesting. The pi. Now, why is this happening? Well, we can actually prove it because cotangent of x plus pi is equal to cosine of x plus pi over sine of x plus pi. Now, what happens with sine and cosine if we are aging pi to the angle? So from this position, it goes to this position. Both ordinate and abscissa retain their absolute value, but change the sine. So basically this is minus cosine of x divided by minus sine of x, which is, again, negative over negative. It's the same thing as cosine x over sine x, which is a cotangent of x. So cotangent x and cotangent of x plus p are exactly the same, which means that the pi is a period. All right, so we've done that. How about pi minus x? That's another little story. First of all, again, from the definition of cotangent as cosine over sine, it's obvious that cotangent is an odd function. It changes the sine of the function. If you change the sine of the argument, why? Because cosine is even function. It doesn't change the sine. So cotangent of minus x equals cosine of minus x over sine of minus x equals cosine of x because cosine is an even function. So cosine of minus x is the same as cosine of x, but the sine is an odd function, so it's minus sine of x. And altogether, it's minus cotangent of x. So, function is changing sine if argument is changing sine, and that's why the graph is supposed to be symmetrical relative to the point zero, zero, zero, right? And it is symmetric, so it's a central symmetry. It will turn it around by 180 degree. It turns onto itself. Now, since we have this property of the cotangent to be an odd function, then let's address this cotangent of pi minus x. What is it equal to? Well, let's change the sine. It's equal to minus since it's an odd function. It's minus cotangent of pi of x minus y, right? We changed the sine of the argument from pi minus x to put x minus pi. Now, since cotangent is a periodic function, we can add pi and we will get minus cotangent of x. So pi minus x is minus cotangent of x. Pi plus x or x plus pi, it's the same as cotangent of x because pi is a period. All right, so basically that's all the functions, that's all the properties I wanted to discuss about cotangent. As far as graph manipulation, obviously if you will have some complicated function which is based on cotangent, like three times cotangent of two times x plus whatever, something like this. So general manipulation with graphs are applicable and obviously you can squeeze, you can stretch, you can shift left, right, upper, down to graph based on these linear transformations of argument and the function itself. And that's all valid and we can do it with this graph as well and we will in some exercises which will be in another lecture or so. So that's it for cotangent for this I would say pariah of trigonometric functions. But anyway, it has exactly the same rights as any other trigonometric function like tangent or sine over there. It just may be less used, but that's a different story. Well, here's what's important probably. Sine and cosine are usually, well, they used to be tabulated and then tangent and cotangent were just calculated based on the ratio, the definition of this. And that's why probably people spent a little bit less time talking about what tangent properties as a function or cotangent properties are. And same story will be with second and cosine. But that's the next lectures. Meanwhile, thanks very much and good luck.