 Hi, I'm Zoran. Welcome to Inizor Education. Continuing talking about different trigonometric functions, today is about second. Well, second is, by definition, one over cosine phi of the angle for any angle. Since cosine is defined for any angle, this is also defined for almost any angle. Obviously for those angles where cosine is not equal to zero. A different definition, absolutely equivalent, based on the unit circle. So if you have an angle phi, now the cosine is obsessive, which is this x coordinate. So in this case, second is defined as one over x. Now obviously, if the angle is between zero and phi over two, in the first quadrant, obsessive is positive, as well as in the fourth quadrant, when an angle is greater than three phi over two. In other case, the second and third quadrant, the obsessive is equal to is negative, and when it's negative, obviously the second is negative as well. Okay, so much about definition. This is a simple thing, and it's basically sufficient to analyze. Or the unit circle doesn't really matter. Okay, so next thing is to talk about properties when the function grows, when the function decreases its values, where it's defined, what's the range, etc. All right. As far as domain of the function, let me start with cosine, the graph of the cosine. Okay, now the cosine goes like this. This is minus one. So, first of all, wherever the cosine is equal to zero, the function is not defined, our second function, because second is again one over cosine. So, the domain is x not equal to phi over two, or minus phi over two, or three phi over two, is generalized as phi over two plus phi times n where n is any integer. Alternatively, you can say that the domain is defined on this interval without edges, on this interval without both ends, etc., which means that x is strictly less than minus phi over two plus pn plus phi over two plus pn where n any integer. So, for n equals to zero, we have from minus phi to phi, which is from here to here, when n is equal to one, that will be phi over two to three phi over two, which means from this to this, and since these are strict inequalities, the points are not included. So, the union of this for all integer ends of this particular interval, if you wish. I mean, you can basically symbolize it as this. This is domain. This is simpler of this. Now, as far as the range, now we have to really investigate how the function grows. So, let me do another graph here. And what I will do, I will analyze how the cosine is changing its values, and then correspondingly, I will invert these values to get into the second. Now, where the cosine is equal to one, obviously, inverted value is one as well. Then the cosine goes down to zero at phi over two. Well, so inverted value would go to infinity, obviously, because this is denominator. It goes to zero, which means the ratio goes to infinity, which means that we have an asymptote at phi over two. It goes like this. Now, symmetrically, if you wish, you can put another asymptote as minus phi over two, and obviously, gain function goes this way, so inverted function goes this way. Well, next. Next, we are jumping over point x equals to phi over two because that's where the function cosine is equal to zero and second is undefined. But what's important is the absolute value of the denominator around this point is zero. It's positive on the left and negative on the right side of the phi over two, which means inverted would be close to infinity or tends to infinity. When we are approaching this point from the left, it would be positive infinity. If we are approaching from the right to the same point, phi over two, we will go to minus infinity. So this asymptote is continuing, and it goes like this. And obviously, a point pi where the cosine is equal to minus one, second is also equal to minus one. Then the situation kind of repeats itself. This is three pi over two. This would be another asymptote. And obviously, as the function increases, in value to zero, the inverted function would decrease to infinity with a minus sign since cosine is negative. So it goes like this. And then it repeats basically this particular thing. So it's two pi. It's one, obviously. And it goes this way to the next asymptote. Similarly, here, I will have minus. So this is a graph of a second. It's inverted relative to the cosine. Wherever the cosine is one for minus one, second is also one for minus one, for one or minus one. And wherever the cosine goes to zero, the inverted second goes to infinity, and it's plus infinity or minus infinity, depending on the sign of the cosine. So what's the range of this function? Well, obviously, the range is everything which is greater or equal than one, and unionized with less than or equal to minus one. In one simple inequality, you can have absolute value of y greater than or equal to one. That's what basically it means. y is greater or equal to one, or less than or equal to minus one. This interval from minus one to one is not in this range. This function cannot take any value from minus one to one. So that's basically how the function behaves, and that's probably the most important part of this lecture. What else? Where this function is equal to one? Well, obviously, at zero and a period of two pi, I can say that y is equal to one, I meant one, I'm not sure what I said. For x is pi n where n is even, and the function is equal to minus one at this point, at pi, three pi, et cetera, where x is equal to pi n where n is odd. Well, I could have written it differently. I can have x is equal to two pi n where n is anything, any integer, and pi times two n plus one where n is any integer. So I'll just express as a formula even or odd, where I can just use the words to specify this. So this is where function is equal to one or minus one. Well, basically, that's it. Now the properties of related to periodicity, asymptotes, yeah, asymptotes are at pi over two plus pi n. So asymptotes are x is equal to pi over two plus pi n where n is any integer. So this, this, this, et cetera. Okay, fine. Now let's talk about some other properties. Almost done with this. Actually, we don't need perhaps a tool. I'll explain it differently. I usually try to specify what would be with these type of values or this type of values or this type of values. Well, these two are very much related. Now, here is very important property of the cosine. The cosine is even function. Cosine of plus x and cosine of minus x are exactly the same. The easiest way to basically, not to remember it, but just to derive this function, the cosine is an absciss, which means that it's x-coordinate. So plus angle or minus angle would go into the same x-coordinate, abscissa. So since cosine is even function, second is also even function. It does not change the sign if argument changes the sign. So this is equal to x minus pi. I don't really need this. Okay, now let's compare this second of pi minus x with second of x. If this is my x, if this is my angle phi, and this is x. Now, pi minus x, it would be if we will go from this position by the same angle. So what happens with abscissa with x-coordinate? If you change this point to this point, if these two angles are congruent, well, obviously abscissa will change the sign. It's very easy to prove the congruence of these two triangles, which means that would be minus second of x. It's always easy to imagine whatever you need with an acute angle and starting from this acute angle, do whatever is necessary to this type of manipulation. So if phi is the acute angle that pi minus x or pi minus phi would be the angle which is on the left side of the y-axis. Now, and if you understand what's this relationship for acute angle, you just transfer it blindly to any other angle because trigonometric functions are defined properly, which means whatever is true for acute angle, this type of relation, it's usually, I'm not usually, it's always actually the same for any kind of an angle. That's why we have defined it this way. It's a proper definition. Now, how about this one? Well, you can do it geometrically as well. Now, if this is phi, then this would be phi plus phi, right? Which means it will also have, these are vertical angles, by the way, because it's a straight line. You add phi to the phi, which means it's a straight line. So, again, triangles are geometrically congruent, and the only thing is different, this x coordinate of this point would be negative. So it's also minus sec, second of x. Now, a different way to prove it would be to use the periodicity. Sec of x plus phi is equal to sec of x plus phi plus phi minus 2 phi, right? 2 phi is a period. So I can add to the argument 2 phi or subtract from the argument 2 phi. We'll have exactly the same value. And this is sec of x minus phi, and I have already determined that this is minus second of x, right? Same thing. So this is kind of a more algebraic way of deriving this thing, and the 1 to 4 was more geometrically. Alright, anything I missed? No, I think that's basically, that's basic properties of sec. I think what's very important actually, as far as remembering, is sec is 1 over cosine. Same thing as you have to remember, the tangent is sine over cosine, and the cotangent is the other way around. You have to remember that second is 1 over cosine, by definition. And the cos second is 1 over sine, that would be the next lecture. It's not too much to remember, but basically I mean it's a definition, so you can't really derive it from anything. Using some logic, there is no logic, just the definition and the words that you have to remember. Okay, that's it for today, and the next lecture will be about cos second, which would end my introduction to trigonometric functions. These are all kind of introductory lectures, because you're just explaining the properties. I'm not solving any equations, proving any theorems or anything like that. These are all trivial properties. Anyway, thank you very much, and good luck.