 Hi, I'm Zor. Welcome to a new Zor education. I would like to continue talking about different trigonometric functions. Today's topic is cosine. In many respects, cosine is a lot of the sine function, which was the previous lecture, and I'll try to be maybe a little bit faster explaining the properties of the cosine, considering you have already familiarized yourself with the sine. Alright, so properties are, first we'll start with definition. Now, just recall that if this is a unit circle on the coordinate plane, and you will take any point A and you will measure an angle phi from positive direction of the x-axis towards this particular direction from O to A. Now, this angle is phi, and for this angle you will define a cosine of the angle phi as an axis of point A. So this is x, this is y. For sine, if you remember it was an originate, but for cosine it's an adjacent side if this is a right triangle. But obviously the definition doesn't depend on the fact that this is the right triangle because the definition of this particular, if you will see this angle from this over there here. So it's almost like full circle just a little bit less. The definition is still the same. It will take the absence of this particular point, which is this in this particular case. So the definition is the x-coordinate of the point which designates the angle we are talking about. Now, usually if we are talking about trigonometric functions, as you remember we measure angles mostly in radians. It doesn't mean that this is an incorrect, no, this is a correct writing. However, when we are talking about function like y is equal to cosine of x, then we are assuming that x is radians. Unless specifically set otherwise. Alright, so that's all about the definition. Now let's talk about how this particular function behaves. So we are talking about x-coordinate of the function of the point based in this angle. So let's start moving. First we will start from the point A which is at angle 0. Angle 0, which means it coincides with this particular point and its co-ordinates are x is equal to 1 and y is equal to 0. So this point corresponds to an angle of 0 radians. Now, what's the x-coordinate of this? Well, obviously that's x, that's 1. So when phi is equal to 0, cosine of phi is equal to 1. Actually, I would prefer to use instead of angular letters like Greek letter phi, I would just use an x which is usually used for the functions. But x means actually an angle in radians. Now let's move our point A over this unit circle, increasing the angle from 0 as it is here to the right angle which is equal to phi over 2 radians. So if x is equal to phi over 2, cosine of x, and this is an x-coordinate, which is this particular point, and the co-ordinates of this point is 0, 1. 0, this is an x-coordinate, and 1 is y-coordinate. So the cosine is equal to 0. Alright, let's move forward this direction. So we're increasing the angle. Now, as point A moves towards this position, which is minus 1, 0, and this is a straight angle which is 180 degrees or pi radians. And as you see, the x-coordinate is equal to minus 1. Next is 3 quarters of a circle. This is the point, its co-ordinates are 0 minus 1. So when x is equal to 3 pi over 2, because this is an angle 3 pi over 2, cosine x is equal to 0. Now, incidentally, we can come to this particular point which is angle 3 pi over 2 counter-clockwise. We can come to this point clockwise with moving this direction, which is a negative direction for an angle. So if the angle is minus pi over 2, it implies exactly the same thing, that the cosine is equal to 0. X is equal to pi, this direction. But we can come to this point moving to a negative direction of the angle measurement clockwise. We will come to exactly the same point if we will move minus pi, right? So if x is equal to minus pi, exactly the same result. And pi over 2, which is this angle, you can move clockwise by minus 3 pi over 2, right? So that's also the same thing, and it follows. And this is obviously if x is equal to minus 2 pi, or if x is equal to 2 pi, that's all the same. 2 pi means the full circle, right? This direction, or minus 2 pi, we're still coming into this point and that's why in all these cases the value would be like this. So let's draw the graph. I'll put this circle here. So this is 1, 0. This is 0, 1. This is minus 1, 0. And this is what? 0 minus 1. Okay, so now we're talking about graph. We start from x is equal to 0, which is this, and 0.1. x is equal to 0, cos n is equal to 1. Now, as x moves towards the increasing the value, which means our angle is increasing counterclockwise, the next stop is pi over 2 where the function is equal to 0. So you will have something like pi over 2 when the function goes to 0. Then the next one at pi is equal to minus 1. Now, as you see, it very much resembles the sign. Then it goes to 3 pi over 2 somewhere here, the function of this and this way. Now, going to the negative side, we will have a very similar one. Minus pi over 2 is 0 again. Then at pi it's equal to minus 1, etc. So the curve itself looks exactly like the sign. Now, the only difference is, if you remember, sign function starts at 0, 0, and then it goes to a maximum of 1 at pi over 2, then 0. So it's just, so this is the sign. So the whole graph is actually shifted. Now, we will investigate what exactly the correlation between sine and cosine and why the graph really looks like it's shifted and it is, but that will be a separate lecture. Right now we are concerned with the blue line, which corresponds to the graph of y is equal to cosine of x. Now, as far as periodicity, now obviously you understand that the function has a period and the period is obviously equals to 2 pi because after the full circle, our angle after 2 pi regions, our circle actually, our point on the circle returns to the same value and that's why it's co-ordinate, subsist angle, coordinate are exactly the same. So 2 pi is a period and you can observe it here. Like from here, for instance, to here, then the function repeats itself or from minus pi to pi, for instance, the function repeats itself. It's supposed to be minus. So this is the periodicity. Now, what we can say about these nodes where function is equal to 0, you see minus pi over 2, pi over 2, 3 pi over 2, etc. So the function is equal to 0 at x equals to pi over 2 plus pi n where n is n in the integer number, positive or negative for 0. If it's 0, it's pi over 2. If it's 1, it's 3 pi over 2. If it's minus 1, it's minus pi over 2, etc. So the function is equal to 0, roots, if you wish, of the equation, cosine of x is equal to 0. Now, where the function is equal to maximum, 1 at 0 and by period by 2 pi as many times as we want. So cosine of x is equal to 1, which is maximum. At x is equal to 2 pi n where n is any integer number. If n is 0, we have 0. If n is 1, it's equal to n. 2 pi, if n is equal to minus 1, we have minus 2 pi, and it repeats by periodicity. And finally, minimum function when the cosine of x is equal to minus 1, minus 1, we have minus pi, pi, etc. So it's at x is equal to pi plus 2 pi n plus the periodicity. So that's maximum and minimum. What else is important? Now, as you see, graph is symmetrical relative to the y axis. So it means what? It means that the function is even, which means that the value of the function of positive argument is exactly the same as the value of the function of a negative argument with the same absolute value. So I would like to say that that the cosine of x is exactly the same is exactly the same as cosine of minus x. Now, well, obviously you can see from the graph, but it's probably better to look at this circle. Again, unit circle. If you will take an angle, which is, let's say, positive, this is 5. And then negative, which is this way, minus 5. The coordinate of this function, at this point, would be, if you will draw the perpendicular here and perpendicular there, it will drop into the same point, because these are equal right triangles, congruent, I should really say, by hypotenuse and an acute angle. Now, if this is not an acute angle, if it's an angle something like this, this is one, and this is another. It's still the same thing, because you can consider the angles which are pi minus this, which is this one. And here also, pi minus this, which is this one. And the same congruent triangles you can consider, and again they're congruent because the hypotenuses are congruent to each other and the angles, acute angles. So from equality or congruence, rather, of these two triangles, this and this, follows that these quantities of one should be equal to quantities of another, which means they are projecting to the same point, and that's why this x coordinate is exactly the same for these two. So it will be minus x and, sorry, it will be the same x. And minus y. Minus y goes this direction, because these are also equal in lengths, but opposite inside, inside. So this is a cosine, this is a cosine, and they correspond to each other. So the function cosine is even and the value of this function for any positive angle is exactly the same as the value of a corresponding negative with the same actual thing. All right, so what else remains to be... All right, now how about we were talking about this for science, how about this? Well, let's go back again to our unit circle. Let's consider first an acute angle, that's easier. So if this is pi, this is pi minus x, sorry, this is x. This is pi minus x. Now, it's, well, if this is pi minus x, it means this is exactly x, right? So these triangles are obviously congruent to each other, again, the angle, the right triangles and the same pathogenesis since it's a radius of the same circle. So these to catch it are equal in lengths, that would be the proper direction. Now, what if it's pi plus x or x plus pi, it doesn't really matter. This is pi and this is x on this side. Same thing. We are projecting this point to exactly the same. So again, the same thing is here. By the way, what's interesting here is how to obtain this from this without considering any any drawings, any unit circle, et cetera, et cetera. Algebraically, rather than geometrically. We already know that cosine is an even function, which means that cosine of pi minus x is equal to cosine of x minus pi, right? I'll just change the sign and according to this it should be the same. Now, we also know that cosine is a periodic function. Which means it's the same as if I will add 2 pi the period to the value of of argument. And what is this? This is exactly cosine of x plus pi which is the same as this. So that's how we basically prove that cosine of pi minus x is the same as cosine of pi plus x, right? x plus pi, whatever. Which is equal to minus cosine. So, this last ecology I could have obtained from the first two just by using the fact that it's an even function and it's a periodic function. Now, once we know the graph we can manipulate this graph using certain known factors about how to manipulate the graph. Now, for instance, you can basically as an example I think I have something like 3 cosine of x over 2 plus p over 4 something like this. How to draw this graph? Well, first you have to change it slightly so you will have the sequence of so it's x plus pi over 2 over 2 right? Common denominator I use 2 but instead of p over 4 I could p over 2 over 2. Now, first I start with cosine of x. Then I have to draw the cosine of x plus p over 2 which means shifting the whole graph by p over 2 to the left. This blue one goes to the left so it will be something like this. Then you have to divide the argument by 2 which means the whole graph would stretch by the factor of 2 so it will be from minus pi minus 2 pi to plus 2 pi more waves will be like this and then you have to multiply it by 3 which means it will stretch vertically by a factor of 2. So that's how you manipulate. So in any case that's basically everything I wanted to talk about cosine. Now these are properties of the cosine by itself. I am not talking about relationship between sine and cosine some transformation formulas etc. So that will be separate lectures. So far this is just about the function why is it with cosine effects. And thank you very much. That's it for today. Goodbye.