 Hi, I'm Zor. Welcome to a new Zor education. This is the lecture number four in the mini series about different inverse trigonometric functions. This is about arc cotangent. Now, before talking about arc cotangent, as usually in case of inverse functions, I have to talk about the original cotangent function. And then we will talk about how to define its inverse. So the definition of the cotangent is traditionally cosine over sine. So let's build the graph of this thing. So first cosine, which is this, cosine is symmetrical relatively to the y-axis. It's an even function, and this is pi over 2, this is pi, and this is 3pi over 2. This is minus pi over 2, this is minus pi, minus 3pi over 2. So that's my cosine. Now sine starts with 0 and goes to the maximum of 1 here, and then goes down, and then goes to minus 1. Here, whenever it's 0, it's minus 1, and it goes to 0 and the plus. So that's the sine. Now let's divide one by another. Well, let's start from 0. Function cotangent at 0 has an asymptote because the denominator is equal to 0. Now both numerator and denominator are positive, which means it's a plus infinity. Now the next point is this one, and that's where the cosine is equal to 0 and sine is equal to 1. This is cosine and this is sine. So the result of the division will be 0, so we will have this. And then I have another asymptote at pi when sine is again is equal to 0. Now cosine is around minus 1, which means starting from this point, the result, the cotangent will be negative because the cosine is negative and sine is positive, so it will go like this. Now obviously it's periodically repeating itself, so I'm not going to draw anything. What's important is we have to choose the domain for the new function cotangent where we can actually find out what's the angle knowing the value of the function. Because if I have all the components, if I have the function cotangent defined everywhere from minus infinity to plus infinity, then we have no way to determine what's the angle if we know the function, so we cannot inverse the function. Because for every value I will have more than one, well actually infinite number of angles cotangent of which is equal to that value. So what I have to do, I have to reduce my domain from all the real numbers to only the place where I know that the function is inversible and in this case the perfect area is from 0 to pi. So let's wipe out everything else. So we will define a brand new function which we will also call cotangent, but it's a new cotangent which is defined only on this interval from 0 to pi, not including the boundary, so obviously because the function is undefined at 0 and at pi. Now what's the range of this function? Well obviously all numbers from minus infinity to plus infinity. Now here on this particular domain with this particular range we can invert the function. We can define what's the inverse function because for every value of the function we have only one unique value of the argument. So from y we can find out x. So in this case we can talk about inverse function called arc cotangent of x. Now its domain would be the same as the range of this function which is y which is from minus infinity to plus infinity. Now its range what's the variance the angle can take if the variance of cotangents such and such. Well from 0 to pi we have already defined this. Now the question is how the function looks like. Well we know that the graphs of the function and its inverse are symmetrical relatively to the angle bisector. So we have to turn our graph around this line. So how would it look? Well these two asymptotes would be turned to the horizontal one. So let's wipe out our sine and cosine which we don't really need anymore and this one. Now let's just draw. So this asymptote would turn into the horizontal asymptote on the level of pi. Now this piece would go into this. This is pi over 2. This is pi over 2. So it would be approximate. So this is a graph of the function arc cotangent. I draw it based on the symmetry between the graph of the function and the graph of the inverse function. I have reduced my original function cotangent to the area from 0 to pi where its monotonically decreasing from plus infinity to minus infinity as the angle goes from 0 to pi not including 0 and pi not including the boundaries. And the inverse function has the domain which corresponds to the range of the original function. The range is all real numbers. And the inverse function has the range which corresponds to the domain of the original function. So that's the definition. And again you can actually express it in words. So what is the angle cotangent of which is x? That's basically the definition of the inverse function. And again from 0 to pi you can always determine what exactly is that particular angle. Well that's it. That's the brief introduction into what function arc cotangent is. I do recommend you to go to Unisor.com website. Read again the notes for this particular lecture. It's really simple. All we really have to understand is the general behavior of the function, its range and its domain. That's basically it. All the problems related to this will be in the future. Don't worry about that for now. So thanks very much and good luck.