 Hi, I'm Zor. Welcome to a new Zor education. This is the last lecture in the mini-series about what values trigonometric functions, in this case it's cosecant, take for all the basic angles. Now, basic angles are those where we can calculate basically the value of trigonometric functions using some algebra and some theorems of geometry. It's all based on two main angles, like 30 degrees and 45 degrees in the right triangle. Everything else is basically derived from this. Let me just remind you very quickly that if you have an angle of 30 degrees or pi over 6, if you wish, in radians, then the opposite calculus is equal to half of the hypotenuse. Very easy to prove, just construct the triangle symmetrical like this. You will have 60 degrees, 60 and 60, so it's equilateral triangle, which means that this side is equal to this side and this is double the calculus. So if this is one, this is one-half. Now, if this is one-half, then we can very easily calculate the second calculus, which is by Pythagorean theorem x squared plus one-half squared is equal to one squared, right? x squared plus one-half squared equals to one squared, which means x squared is equal to 3 quarters and x is equal to square root of 3 over 2. So let's just put it in, square root of 3 over 2. I never remember anything like this. I always remember just one-half everything else can be derived using the Pythagorean theorem. And another example I was saying about the angle of 45 degrees also very easily calculated. If this is 45 degrees, this is also 45 degrees. So this is one and this is x and this is x, so x squared plus x squared is equal to one. x squared is equal to one-half and x is equal to one over square root of 2, which is square root of 2 over 2. So let's just remember this. We didn't really memorize it, we derived it. That's always my preferable way of doing things. Now, knowing that all other angles which are kind of related to this like 120 degrees or 270 degrees, 150 degrees, whatever else, or negative 120 degrees, all these angles, which we also call basic, are very much related to these ones. And I will calculate the value of cosecant for all the angles just based on whatever I have here and certain considerations of symmetry I have introduced when I talked about basic angles. So we go back to our unit circle. Let's mark our angles like this. This is 30 degrees, this is 45, this is 60. Now, why do they put 60? Well, for obvious reasons. If that is 30, then this is 60. So it's all easy to calculate. Drop down the perpendicular. Now, this is a unit circle which means the k-potanus in each of these three right triangles is always equal to 1. Now, the smallest cactus, now if this is 60, then this is 30. So this is 1 half. Let me put it here. This is 1 half. This is square root of 2 over 2 and this is square root of 3 over 2. So the coordinate of this point is, this is 30 degrees. You don't need this anymore, which is pi over 6. And the coordinates of this point are square root of 3 over 2 comma 1 half. Because the coordinate is equal to this is 30 degrees. This is 45 and this is 60. So, the coordinate is k-potanus. Now, this is 45 degrees, which is pi over 4. And the coordinates are square root of 2 over 2, square root of 2 over 2. Both cadets are equal to each other. Finally, 60 degrees. Now, the abscissa is 5 over 3. This is 1 half and the coordinate is square root of 3 over 2. So you know about this. Now, what is a cosecant? A cosecant is 1 over cosine, 1 over sine. Yeah, 1 over sine. A cosecant is 1 over sine. So, what is the sine? What is the coordinate in the unit circle? For each angle, I get this point, which makes this angle from the positive direction of the x counterclockwise. So, this is 30 degrees, 45 and 60. So, if I will take an coordinate, which is 1 half, in this case, square root of 2 over 2 and square root of 3 over 2. If I will invert it, like in this case it would be 2, in this case it would be 2 over square root of 2, which is square root of 2. And in this case it would be 2 over square root of 3, which is 2 square root of 3 over 3. So, these are cosecants of these angles. But we already spoke about that before. Now, we are talking about other angles, which are outside of the first quadrant on the coordinate plane. Okay, so I have a list of all these angles. So, let's just do it one by one. It would be easy for me if I would do this. These are my angles. Okay, the angle number one is this one. It's 120 degree, which is 2 pi over 3. Now, 120 degree, and this is 60 degree. Look at this, 90 plus 30, 90 minus 30. So, these two angles are completely symmetrical relatively to the y-axis. Now, if these angles are symmetrical, I've proved in the lecture about basic angles that these two points are also symmetrical, which means they are on the same perpendicular to the y-axis. And their x-coordinates, abscissas, are opposite in sign, but the same in absolute value. That's what makes these symmetrical. So, these are two equal congruent actually segments, and this is one perpendicular. They share the perpendicular to the y-axis. Okay, now, if that's true, then what's the co-ordinate at this point? Well, again, the coordinate is retained, which is this one, but abscissa is opposite in sign, so it's this. So, what's the value of cosecant, which is 1 over sine? Sine is an originate. Originate is retained. So, it's 2 over square root of 3 or 2 square root of 3 divided by 3. That's my cosecant. Next, this is 135 degrees or 3 pi over 4. Now, coordinates of this are this point is obviously symmetrical to this one. 135 is 90 plus 45. 45 is 90 minus 45. So, these points are symmetrical, which means my coordinates are the same, my abscissa is changing the sign. So, it's minus square root of 2 over 2, square root of 2 over 2. So, my cosecant, which is 1 over sine, which is 1 over the originate, 2 over square root of 2, which is square root of 2. Next is 150 degrees, which is 5 pi over 6. Symmetry is obviously with this point, because this is 90 minus 60 and this is 90 plus 60. So, I retain my originate, I change the sign of abscissa, so it's minus square root of 3 over 2, 1 half. And what's the cosecant, which is 1 over sine, which is 1 over originate, which is 2. Next is pi. Pi has coordinates minus 1, 0. And the cosecant is obviously undefined, because we cannot divide by 0. So, the cosecant doesn't exist for a 180 degree or pi angle. Next. Next we go this direction. Just for fun, you can go forward and increasing the angle, or you can go in the opposite and negative direction and basically come to the same point, this or this or any other, using the negative. So, I choose just for diversity to use negative angle. So, this is minus 30 degrees or minus pi over 6. And obvious symmetry is with plus pi over 6, right? These two angles are symmetrical relative to the horizontal axis, which means they are projecting into the same, which is abscissa, which is square root of 3 over 2, but their ordinates are opposite in sine but equal in absolute value. So, in this case, it would be minus 1 half. Now, what would be then the cosecant of this particular angle? Well, if sine is minus 1 half, cosecant is reverse, so it would be minus 2. Okay, next. Next is minus 45 degrees or minus pi over 4. Symmetry is with this point, plus 45. So, I'm retaining abscissa, which is square root of 2 over 2. I'm changing the sine of the ordinate and my cosecant would be minus 2 over square root of 2, which is minus square root of 2. Next is minus 60 degrees, which is minus pi over 3. The coordinates are symmetrical with this point, plus 60. So, I retained the ordinate, but I changed the sine... Sorry, I retained the abscissa, I changed the sine of the ordinate. So, it's 1 half minus square root of 3 over 2. So, I'm interested in the ordinate. Inverting this would be minus 2 over square root of 3, multiplied by square root of 3. So, it's minus 2 square root of 3 divided by 3. Minus pi over 2, minus 90 degrees. The coordinate of this point is immediately 0 minus 1. So, knowing coordinates, I can invert my ordinate and that would be my invert of a sine. That's my cosine, which is equal to minus 5. Next is this, which is minus 120 degrees, which is minus pi over 3. Coordinates are obviously symmetrical with this guy. So, I retained my abscissa, my ordinate gets changed. That's coordinates. And therefore, my cosecant is minus 2 square root of 3 over 3. This is minus 135 degrees, which is minus 3 pi over 4. My symmetry is with this guy, which is minus square root of 2 over 2. I changed the sine of the ordinate, which is also minus square root of 2 over 2. And my sine will be minus square root of 2. Inverse, this one. Next is this, which is minus 150 degrees, which is minus 5 pi over 6. Symmetrical to this point, minus square root of 3 over 2 minus 1 half. I inverted the ordinate. So, my cosecant is invert of minus 1 half, which is minus 2. And finally, I've got to this point, which is minus pi, which is the same as plus pi. And the coordinates are minus 1, 0, so you cannot divide by 0, so it's undefined. So, these are the values of cosecant for all the different basic angles. Again, I started from the very basic angles, which are from 0 to 90, like 30 and 45 basically, then derived for 60, and then used symmetry to derive for anything else. I do suggest you to go back to unizord.com, go to the notes of this lecture, where all these examples are written, and there are answers, I think, as well. So, try to do exactly like what I just did yourself, and check if you have the same answers. It really helps you to basically better understand how all these manipulations are done, and it will be almost like automatically for you. That's what I do suggest. That's it for today, and this is the end of this mini-series of lectures about what are the various of different trigonometric functions for different basic angles, not only within the first quadrant, but in all four quadrants of the coordinate plane. Thank you very much, and good luck.