 Hi, I'm Zor. Welcome to Inizor Education. Today we will talk about trigonometric function of sine. Actually, my plan is to explain all the trigonometric functions, but the first one for sine I will probably do in more details. So in any way, how can we examine the function? Well, basically, we were talking about different properties of the functions, domain, range, graph, properties, etc. So basically, that's what I'm going to analyze right now. Alright, so sine of x, y is equal to sine of x. First of all, let's talk about what is x. We were talking about sine as a function which is defined for angles. Now, the definition of sine was, if you remember, we had a unit circle and for any angle which we start from positive direction of the x-axis and move counterclockwise to whatever the angle value actually is, we get the point A with coordinates x, y and sine of this angle f is, by definition, is y-coordinate. So the length of this catatose in this right triangle. Don't forget that this is a unit circle, so hypotenuse is equal to 1. That's why the ratio is always equal to y-coordinate. Now, what is the angle phi in this particular case? How is it measured? Well, we know that angles are measured in either degrees when the whole circle is divided into 360 degrees or in radians and the radian is basically an angle which corresponds to the arc which is equal to the radius in length. And I was also talking about radians as being a preferable measure for angles in trigonometric functions when we are using the function context. Well, so let me just tell our front. So whenever we see something like this, what we actually mean is that x is a numerical value of some angle in radians. So sine of 1, let's say, is a sine of an angle of one radian approximately like this. Now, if we are talking about sine of 2 pi, it means the sine of an angle of 2 pi radians. 2 pi radians obviously is a full angle completely, which is equal to 360 degrees. So x is an angle in radians. Now, also I mentioned that this particular angle can be measured either in a positive direction. So whenever x is positive, it means we are starting from here and move whatever number of radians counterclockwise. However, x can be negative as well. And the angle in this case is we are moving clockwise towards this direction from the positive direction of x. So the sine of minus 1 is actually something like this. This is angle of minus 1 radian. And the sine would be the y-coordinate of this particular point, which is negative, by the way, because it goes down. All right? So this is as far as the definition is concerned. Now let's investigate how this sine behaves as we are changing an argument. And for this, I will draw a graphical representation of this function. So let's start from the beginning when x is equal to 0, which means we are not moving from this direction. And the point A actually takes this position, which is 1, 0. This is 1, and vertical position is 0. So what's the sine of the angle of 0 radians? Well, that's the y-coordinate, which is 0. So whenever x is equal to 0, y is equal to 0. So we start from this particular point when we are drawing the graph of our function. Now as x is increasing, point A moves along this circle towards the uppermost position when the coordinates are x is 0, and y is 1. And the angle in this case is equal to, so what's this particular angle? In degrees, we kind of fall comfortable, this is 90 degrees, because it's one quarter of the full circle, which is 360, right? Now in radians, the whole circle, the full circle angle is 2 pi radians, because circumference of the circle equal to 2 pi r. So how many pieces of lengths are in the full circle? 2 pi. So the full circle is 2 pi, so a quarter is 2 pi divided by 4, which is pi over 2. So whenever our x is equal to pi over 2, y takes its maximum value of 1. Now how's it moving in between? Well, it's not easy right now to say, well, maybe it moves like this, maybe it moves like this, well, maybe it moves even by straight line. Well, a more precise analysis shows that the shape of this curve is this, more than that actually. If you draw a tangential line to the point 0, it will be exactly 45 degrees to the bisector of this sample. So let me just draw it a little bit more accurately, so you will just see what I mean. So let me start with bisector of this line. So the graph would look like, so this would be a p over 2 line. So that's how it grows. From 0 at x is equal to 0 to 1 when x is equal to over pi over 2. Next, as we move another quarter of a circle, which means we are moving here, what happens with our y coordinate? Well, it goes back down to 0, right? In this particular point, it's minus 1, 0. So from 1, it diminishes down to 0, my y coordinate. And that's why the graph goes this way. What happens next? Next, my y coordinate is going to continue through 0, going down to minus 1. The coordinate of this point is x is equal to 0, y is equal to minus 1. So we go through 0 to minus 1. And I'm not going to draw it much further because I don't have space, but I'll just indicate it here that somewhere along the line at this point, which is this point is x is equal to 3 pi over 2. This is x is equal to pi. This is x is equal to pi over 2. And this is x is equal to 0, right? So at this point, it's equal to minus 1. And then it increases again back to 0. And that's where the full circle actually is ending. Well, at the same time, as you understand, I can go to negative direction of the angle. So what if x is going to to this point, but not through the positive counterclockwise direction when it's equal to 3 pi over 2, but to a negative point, which is minus pi over 2. So it's exactly the same point, right? Which means it should be exactly the same, which is minus 1, right? So whenever my point is here about the same distance, minus pi over 2. I also have to have the same minus 1, which means my graph should go the same way, the same way to minus 1 here at 3 pi over 2 and here at minus pi over 2. And then after that, if I go even further to a negative side, it would be x is equal to minus pi. Then it increases, my y coordinate is increases to 0, right? So the graph will go like this at point minus pi. And again, it repeats itself, basically. So all I'm saying is that you will have this waving function, which is changing from the value of 0 to the maximum of 1, then goes back to 0, then to the minimum of minus 1, then goes back to 0, etc., etc., is I'm circling round and round and round. Obviously, this is a periodic function with a period equal to the full circle, which is 2 pi. So that's what I wanted to talk about right now about behavior of this function, all right? After we have established general behavior of the function, let's just very quickly summarize the properties of the function. So domain is x can take any value because whatever the value is, positive value, it means it's the angle of regions which I have to move from this direction counterclockwise, whatever number of times. Let's say if my angle is 16 pi, for instance. Well, then it means I have to circle around the center eight times, 2 pi each time, to get 16 pi's, and I will still go to this point. And my sign will be exactly 0 as it was when x is equal to 0. So the function is periodical. The period is equal to 2 pi because the values are repeated itself after 2 pi. The range is from minus 1 to 1. So I can say that my domain is from minus infinity to plus infinity. My range is from minus 1 to plus 1. Here I have to put equal, less than or equal because at certain points it's equal to 1. Now, let's just fix certain specific points. You see this is the point where the function is equal to 0 and it repeats after pi. So my function is equal to 0, y is equal to 0, 4, x is equal to pi times n, where n is any integer number, negative integer, positive integer, or 0. By the way, the fact that it repeats the value 0 doesn't mean that after each, after pi doesn't mean that the pi is a period. Pi is not a period. 2 pi is a period. But 0 does repeat itself because the function has this particular shape. So within the period it actually takes the value of 0 more than once. It's not a monotonous function during the period. What else can be said? Well, when is the function equal to 1? Well, y is equal to 1 when x is equal to pi over 2 plus period 2 pi and the period can be multiplied any number of times. And again, n can be positive or negative. So if n is negative, then 1 would be over there. And also it will be somewhere there. When n is equal to 1, it will be what? 5 pi over 2. It will be over there. If n is equal to minus 1, it will be minus 3 pi over 2. It's somewhere here. And if n is equal to 0, we will have just pi over 2, which is here. So all these are points where function is equal to 1. Now, when is the function equal to minus 1? Well, minus p over 2, x is equal to minus p over 2. And again, plus pi n because 2 pi is plus 2 pi n because 2 pi is a period. What other interesting properties? Let me go back to our circle. And I will probably have to redraw it again because it's too many different things are already there. So this is our circle. Let's take the same angle here and here. So these are the same angles. And let's compare this is x1 and this is x2. These are two angles. So x1 is a certain number of radians. This particular angle is x2 is this particular angle. Now, if these two acute angles are congruent, then let's say this is point a1, this is point a2. This is o, this is point x1, this is point x2. So angle x1 o a1 congruent to x2 o a2. Now, what does it mean from the sine position? Well, sine is an origin, right? But ordinances are the same. In both cases, the origin of this point is this and the origin of this point is also this. So it looks like sine of x1 should be equal to sine of x2. Now, can it be proven? Well, very easily, obviously. I mean, if you compare these two triangles, they have the same hypotenuses because it's radiuses of this circle, they're both equal to one. And since these angles are equal, and these angles are 90 degrees, so these angles will be equal as well because it's 90 degree minus x1. So we have, and this is common. Actually, these are congruent triangles because they have a side angle and a side angle and a side. So this is common anyway. So, well, actually, I shouldn't really say it's common because in theory, I can project, yes, forget about this common thing. For instance, is it possible that they project the different points? This project this and this projects into this point? Is it possible? Is it possible? Well, the answer is no, because since these are projections between these are perpendicular, then these are two right triangles and right triangles with the same hypotenuse and the same acute angle are congruent, which means these two kind of should be of the same length. And that's why the projection is actually to the same point. And this is indeed a common, a common side. Yeah, I shouldn't really assume this is common. Alright, so being as it may, we have proven that these two points have the same origin and that's why this is a true statement. Now, how can I express x2 angle if I know x1? Well, obviously, x2 is equal to pi minus x1, right? Because this is pi and this is minus x1. These two angles are congruent. So, I can rewrite this instead of x2, I can put pi minus x1. And in general, I can say that sine of pi minus x equals to sine of x. So, for acute angles, it's obvious. Well, what if angles are not acute? Well, if you really just move it further, x1 will be moved on this side, x2, therefore, will be on this side, and it will still be the same. So, even if it's of two angles, it's still exactly the same thing. Right? Let me just draw a picture in this particular case. So, my point is that this particular equality is an equality for any angle. So, what if my angle is of two? So, this is my x. And I'm saying that 100 and 80 minus x, I should go this way. So, from 100 and 80, this I should go back by the same angle. So, it would be this. So, it's still exactly the same thing. And this is pi minus x. And exactly the same would be even if x is greater than pi, even if it's in this particular territory, in which case it's just a negative. You can come to the same point through a negative direction. So, no matter how we move, this equation would be held because the ordinance of the points would be exactly the same. This is ordinance. Okay. How about sin of pi plus x? Here is a trick. Since the function is periodic, it's exactly the same as sin of pi plus x minus 2 pi. Right? If I will subtract 2 pi or add 2 pi, 2 pi is a period. So, the function would be exactly the same way, which is equal to sin of x minus pi. x minus 2 pi plus pi. So, it's x minus pi. Now, what's the difference between pi minus x and x minus pi? Well, the difference is a sin. So, whatever the angle is, if this is pi minus x, this is x minus pi. Right? But if I'm moving to a different direction by the same angle, instead of this, I go to this. My ordinate would be the same in absolute value, but different in sin. Right? Because here is a positive ordinate and here is negative. So, if these two angles are equal as angles, they're congruent, then these two also are equal in lengths, but they are opposite in sines. So, this is equal to minus sin of pi minus x. Right? So, if I change the sin of the argument, the function changes the sin. Now, sin of pi minus x, we already said what it is, right? So, this is minus sin x. So, here is basically two final statements. Sin of pi minus x is the same as sin of x. Sin of pi plus x is the same as minus sin x. And incidentally, we actually talked about what happens if I would change the sin of the argument. And as I was just indicating, if I change the sin of the argument, my function changes the sin, because instead of this direction, I go to this direction by the same angle, and obviously, ordinate will be the same in absolute value, but opposite sin. Now, what is this? This is an indication that our sin is an odd function. This is the definition of the odd function. Effort minus x minus x is equal to minus effort x. And incidentally, definition of the even function is this. So, what we are saying is that our sin is an odd function. An odd function, if you remember, had the graph which is symmetrical relatively to the center of coordinate. And what I was just drawing was something like this, which is obviously symmetrical relative to this particular point, the beginning of the coordinate. This is the graph which I was just drawing. Okay. So, function, we talked about the main. We talked about range of the function. We talked about the fact that the function is odd. We talked about these two couple of formulas, sin of pi minus x and sin of pi plus x, how it can be transformed. Let's just draw, as an exercise, one particular graph of some function which is a little bit more complicated than just plain sin x. We know how the graph is supposed to be transformed if we transform some, in some elementary way, if we transform an argument in the function. So, what I would like to suggest right now is to graph the function 3 sin of x over 2 plus pi over 4. Let's draw this particular graph of this particular function. So, obviously, it's based on the function y is equal to sin of x. Now, what's the transformations which we really should do? What is the transformation with an argument? Well, we have kind of a complicated transformation. We divide it and then we add it, which is not exactly the proper way. What I would like to express it is 3 sin of x plus pi over 2 divided by 2. That's easy, because first we add something to the x, then we divide the argument by 2. So, let's just do one by one. We start with plain sin. So, what's the plain sin? It's this. Now, I draw it on minus p, p interval. What's important for any periodic function, you can draw it not only on an interval from, let's say, 0 to 2 pi, because 2 pi is a period, but you can draw it on any interval which has the length of a period and then repeat itself. So, in this case, the period is 2 pi, but I decided to draw it from minus pi to pi instead of from 0 to 2 pi because it repeats anyway. So, this is the beginning. Now, first what we do, we do this, x plus pi over 2. So, I would like to draw a graph y is equal to x, sorry, sin of x plus pi over 2. So, what happens with this particular graph? Well, you know that if I am adding something like pi over 2 to x, the graph is supposed to be shifted to the left by pi over 2. So, let's shift it to the left. What happens? Well, that's what happens. When the blue graph is shifted to the left by pi over 2, the red one comes up, right? Next. Next, I will divide argument by 2. Now, what does it mean when we divide argument by 2? The graph is stretching horizontally from the center both directions by 2. So, the next one would be, so instead of being equal to 0 at pi over 2, it will be equal to 0 at pi and correspondingly here. So, the new graph will look like this and here it will reach 0 to minus pi. So, the graph will be stretched. Finally, if I am multiplying, so this is y is equal to sin of x plus pi over 2 divided by 2. And next one, we are multiplying it by 3. Now, what if we multiply it by 3? Well, it means that it's stretching vertically by a factor of 3. So, instead of maximum being 1 and minimum being minus 1, we will have maximum 3. So, it will be something like this. So, it will be the same style wave, but the wave will be shifted left, right, stretched horizontally or stretched vertically, but the general shape will remain. So, whenever you see function like this, where x is transformed through some kind of linear transformation and the function itself is transformed through some kind of linear transformation, it's the same style of graph, but it will be stretched, shifted or whatever. Well, that basically completes my story about function y equal to sin of x. So, don't forget that whenever you have x as a number, it means an angle in radians. Well, and everything else basically goes like in every other function. You have a domain which is everything. You have a range which is from minus 1 to 1. You have maximum, minimum, all the properties. And you might, yes, periodicity, very important to pi is periodicity. And a couple of formulas which I said about sin of x plus pi or x minus pi whatever. And the fact that the function is odd also very important. Symmetrical relative to the center. That's summary of whatever I said. That's it. I will try to spend some time for every trigonometric function and draw graphs. Probably the graphs are the most important just to show you how the function behaves basically. So, that's it for today. Thank you very much and good luck.