 Hi, I'm Zor. Welcome to a new Zor education. This is the fourth lecture in a series of lectures about what values trigonometric functions take at basic angles. Now, the basic angles are not only your familiar 30, 45, and 60, but also some derived basically from these all around the unit circle. And the unit circle is exactly the tool which I'm going to use to calculate the values of the basic functions of the trigonometric functions at basic angles. Why? Well, because again, there are many functions, many basic angles, and I'm not going to memorize everything. I'm trying to memorize the minimum information and then derive based on some logic all other values. And that's exactly which I'm going to demonstrate with cotangent. As I already did for sine and cosine and tangent in the previous lectures. So this is the lecture number four, cotangent. And to calculate the values of cotangent at the basic angles, which I have just presented here, I need the unit circle. I need to represent these angles as points on the unit circle, find their coordinates, and then use the definition of the cotangent, which is abscissa over coordinate, x-coordinate divided by y-coordinate of the point which represents the end. So let's do it. And how can I derive basically all the values without memorizing everything? Now, I do have to remember something. And the only thing I remember is that in the right triangle, if the angle is, q-tangle is 30 degrees, then the opposite casualties is half of the hypotenuse. So this is the unit circle. So if this is my point, which has 30 degrees, I know that this particular casualties is equal to half of the hypotenuse. Hypotenuse is 1, so this is 1 half. So this one is from the Pythagorean theorem, x-square plus 1 half square is equal to 1 square, right? x-square plus 1 half square equals to 1 square. So this is 1 quarter. So this is 3 quarters. x is equal to square root of 3 over 2. So the coordinates of this point are abscissa is equal to square root of 3 over 2, and coordinate is 1 half. Now, this is an angle of pi over 6, which is 30 degrees. So I've got that. Now, at 45 degrees, I have both casualties the same, because this is 45 degrees and this is 45 degrees, right? So in this triangle, in this right triangle, my both casualties are the same. And to find their value is just to use Pythagorean theorem like this. This is x-square, and this is x-square equals to 1 square. 2x-square is equal to 1. x is equal to square root of 1 half, which is 1 square root of 2 over 2, which means that coordinates of this point are square root of 2 over 2, square root of 2 over 2. This is pi over 4, which is 45 degrees. And the third basic angle in the first quadrant is 60 degrees. And if you will take a look at this triangle, this triangle, it's exactly the same as this triangle. One angle is 30 degrees and another is 60. In this case, this is 60 and this is 30. Hypotenoses are the same. So the same hypotenuse, the same angle, so it's equal triangles, congruent, I should say. Contemporary terminology. All right. So the only difference is this is small which is an originate and the bigger is abscissa. In this case, it's twice worse. The smaller one is abscissa, which is 1 half, and the bigger one is originate, which is square root of 3 over 2. Now, equipped with this, now this is pi over 3, which is 60 degrees. Now equipped with this knowledge, which I did not really memorize, I memorized only one thing that this particular pattern is equal to 1 half. Everything else is derived. So knowing this, I can start calculating the value of the cotangent in this particular case for other basic angles in other quadrants. So now let's go. 2 pi over 3, 2 pi over 3, which is 120 degrees. Now, as you see, this is 90 plus 30, gives me 120. This is 90 minus 30, gives me 60. So these angles are symmetrical. Now, the symmetry of the angles results basically, it implies the symmetry of these endpoints as was proven in the lecture, which was about what are the basic angles in the very beginning of the course of trigonometry, which I had. So these points are symmetrical relative to the y-axis. Now, what does it mean? It means they project onto the same originate. So the originate are the same. And as far as their abscissa is this and this. They are opposite in sign, but the same in absolute value because of the symmetry. So I will retain in this particular coordinate of this point to get to this one. I will retain an originate, and I will change abscissa. Now, what's the cotangent? By definition, cotangent is abscissa divided by originate. So let's do it. Minus 1 half divided by square root of 3 over 2 gives me minus 1 over square root of 3 or minus square root of 3 over 3. That's my cotangent for 2 pi over 3. Well, let's move on. This is 3 pi over 4. 3 pi over 4, which is 135 degrees. Now, same logic. 90 plus 45 gives me 135. 90 minus 45 gives me 45. So these points are symmetrical. The angles are symmetrical. Therefore, these points are symmetrical. And the symmetry gives me the same originate and opposite abscissa. So it's minus square root of 2 over 2 and square root of 2 over 2. And obviously, the cotangent, which is abscissa over originate, gives me minus 1. Next, here is 5 pi over 6, which is 150 degrees. Now, what's this? 150 is 90 plus 60. Now, 30 is 90 minus 60. So these points are symmetrical. So I retained the originate. But I changed the value of the abscissa. Now, the ratio of abscissa over originate will be minus square root of 3. Now, this next is pi. Now, pi has coordinates minus 1, 0. Now, I cannot divide by 0. So the cotangent is not defined at this critical point. Next, minus pi over 6, which is minus 30 degrees. And what is the symmetry in this case? 30 plus or 30 minus? So these two are symmetrical. Now, but these are points symmetrical relatively to the x-axis, which means abscissa will be the same. They are projecting to the same point. But opposite ordnance. So this is the same. So it's square root of 3 over 2, but opposite minus 1 half ordnance. And what's the cotangent, which is the ratio of abscissa over originate? It's minus square root of 3. Next, minus pi over 4, which is this. Minus pi over 4, which is minus 45 degrees, which is symmetrical to plus 45 degrees. So the coordinates are, I retain my abscissa, which is square root of 2 over 2. I change the ordnance to an opposite sign. And the ratio of cis over ordnance would be minus 1. That's the value of the cotangent. Next is minus pi over 3, minus pi over 3, which is minus 60 degrees. Obvious symmetry with this guy, this plus 60 degrees, retain abscissa, 1 half, change ordnance, square root of 3 over 2 is a minus sign. And the ratio is minus 1 over minus square root of 3 is the same as square root of 3 over 3. Next is minus pi over 2, which is this one. It's minus pi over 2, which is minus 90 degrees. Now the coordinates at this point, we don't really have to think about. 0 is abscissa, and minus 1 is originate. So the ratio of abscissa to ordnance would be 0. OK, 2 pi over 3, minus 2 pi over 3, which is minus 120 degrees. Obvious symmetry with plus 2 pi over 3. So let's take a look at this one, retain abscissa, and change the ordnance to an opposite. So it's minus 1 half, minus square root of 3 over 2. Now, what's the ratio of cis over ordnance? Well, both are negatives, so the result is positive. It's 1 over square root of 3, or square root of 3 over 3. Next is minus 3 pi over 4, which is minus 135 degrees. Obvious symmetry with plus 3 pi over 4. That's the wrong one, this one. With a plus 3 pi over 4, so I'll change the coordinate of this thing. I retain the abscissa, minus square root of 2 over 2, and I change the ordnance to the opposite, which is also minus square root of 2 over 2. So the ratio would be 1. Abscissa over ordnance, so cotangent is equal to 1, 4 minus 3 pi over 4, minus 5 pi, minus 5 pi over 6, which is minus 150 degrees, symmetrical to plus 150 degrees. So I will take these guys and retain minus square root of 3 over 2, abscissa, change the ordnance to an opposite, divide one by another, and I will have square root of 3. That's the cotangent. And the last 1 minus pi, which is this point, and that's the same thing as plus pi. And since we cannot divide by 0, its cotangent is undefined. All right, so that was the cotangent for basic angles around the circle. Again, remember that you really can derive all these values based on very few facts which you might actually remember, like the calculus opposite to the 30 degrees angle equals to capital H. But everything else is derived from the considerations of symmetry and the definition of the cotangent, which is cosine divided by sine or abscissa divided by ordnance. So all you need is co-ordinates of all these basic points which represent basic angles, and these co-ordinates are derived based on symmetry. I just suggest you to do it again just by yourself and check the results. Thanks very much.