 Hi, I'm Zor. Welcome to a new Zor education. We will talk today about tangent function. Well, as usually, let's start with definition, which we have talked about once. Let me just remind before going into the properties of the function. Let's define it again. Now, the classical definition of function, the tangent of the angle is sin of this angle divided by cosine of this thing. Now, alternatively, you can always resort to unit circle and basically, well, based on this definition, of course, you can say that if this is angle phi and it goes from the horizontal direction of the O x towards our point counterclockwise. That's the positive direction of the angle by angle phi. And this point has coordinates x, y. Point belongs to the unit circle. Then tangent of phi is equal to y over x, y over x. I don't want to use terminology like opposite-catchitus ratio of the opposite-catchitus towards the adjacent, just because we are not talking only about acute angle phi. It can be something like this. This can be fine. So the point A is here. And the definition still holds, basically. It's the ordinate to abscissa, the y-coordinate divided by an x-coordinate. So whatever the definition is, more classical if you wish or the one which we can use which contains the unit circle, the properties are exactly the same, obviously. Now, the first and most important property of the function looking at this particular definition comes from the fact that there is something in the denominator. Always if there is something in the denominator you have to be very worried about something, which is what? Obviously when the denominator is equal to zero. So when this denominator equals to zero, function is undefined. Okay, not only the function is undefined, but also every point where cosine is equal to zero, the function must actually have an asymptote because at the top you have something which is which has a restricted value. It's no less than minus one and no greater than one. So it's a bounded values. But on the bottom you have in the denominator something which goes to zero, which means the whole ratio should go to infinity. Negative infinity or positive infinity is the story, but nevertheless it goes to infinity, which means in all those points where cosine is equal to zero you will have asymptotes. Now let's basically analyze the behavior of the function tangent as defined in this particular way and we will base this on the functions sine and cosine which we have already used before. So let's do it graphically. We can do it in many different ways but I'm just suggesting to use it graphically. So let's go from zero to two pi because this is basically a period if we will analyze the function on this particular interval it means it will repeat it everywhere else. So this is pi. This is three pi over two. This is pi over two. So first let's look at the sine on the top. So y is equal to tangent of x which is equal to sine of x over cosine of x. By definition there's nothing to talk about this. So first let's draw the sine. Sine would be something like this. Now let's draw a cosine. Cosine on the same zero to two pi would look like this. Okay. Now let's draw the tangent which is basically a ratio of this over this. Well, zero. Now that's where our cosine is equal to one. And sine is equal to zero. So the ratio will be zero. So we will have this point. Now moving to the right my numerator is increasing. My denominator is decreasing. Which means my ratio should increase. This is increasing. This is decreasing. So my ratio is increasing. Now at the point where they are equal to each other which is this one which happened to be pi over four by the way which is 45 degrees. The ratio will be equal to one. And then so let me put this with O. O. And then as I approach pi over two my sine is increasing to one but my cosine decreasing down to zero. Which means what? The function tangent which is a ratio should really go to infinity. And positive infinity by the way because both numerator and denominator are equal to are positive. So as I approach pi over two from the left my ratio should go to plus infinity. So this is asymptote and my graph would look like this I would say. It increases to plus infinity. Now as soon as we jump over pi over two my numerator is still positive my denominator is negative but also very very small which means it's also when I approach pi over two from the right I'm also supposed to increase in the absolute value but the sine would be negative which means on this side of the asymptote I will have a negative infinity. Still infinity because this is still almost zero and this is a bounded area about one but negative. So it goes like this. Now at this particular point right in the middle when they are equal in absolute value but opposite in sine so I will have minus one and then as I approach pi my sine on the top would be equal to zero my cosine would be negative minus one so the ratio would be something like this. After that we are crossing to the positive side because both numerator and denominator are negative so negative divided by negative would be positive. Now so at this particular in between right in the middle it will reach one again but whenever I reach three pi over two I will also have the same story. My denominator goes to zero my numerator is about minus one. So the ratio would be in absolute value and positive in sine because both of them are negative so something like this. What happens after it? Again, similarly to this they are of different sine right now so the ratio would still be infinite in absolute value but negative in sine and finally I will reach zero at this point. So this is my graph on a measurement from zero to two pi. Now what happens after this? Well obviously or before that obviously if I have something like here minus pi over two I will also have an asymptote and the graph would continue this way and so here another asymptote and the graph would continue this way. So basically the graph is also as you see repetitive it has a period and this is a pattern which repeats itself so from zero to pi over two it grows to plus infinity from pi over two to pi it goes from minus infinity to zero then continues to plus infinity and then when we cross over three pi over two again from minus infinity to zero at two pi etc. Now what's interesting is that although periodicity of the sine the red one is two pi and periodicity of the cosine the purple one is also two pi but the periodicity of their ratio repeats itself from minus pi over two to pi over two this particular segment which has the lines of pi not two pi one pi it repeats itself so that's the property so the period of the tangent is pi not two pi obviously two pi is also a period because if the function repeats it's very after pi the function the smallest period obviously so the smallest period in this case is pi now can we algebraically define derived this property that the tangent is a function of a period pi well yes we can let's just remember the following if you take sine of x plus pi so if you add pi which is a straight angle and the sine is an ordinate your ordinate will change the sine but will be the same in absolute value from here you add pi which means you go here so if that was your point in the beginning then it goes to a prime so whatever ordinate was positive here would be negative here so sine of x the sine of x plus pi equals to minus sine of x now how about cosine of x plus pi the same picture cosine is an x coordinate of the point so if it was positive for an x it would be negative for x plus pi so it's also minus so if you will divide one over another you will see that tangent of x plus pi in all those points x where tangent exists of course equals to tangent x which proves that the pi is a period now so we basically have proven this particular property and there is another property which I use usually mentioned as well and we can derive it here as well tangent pi minus x one more thing I forgot to mention before doing this tangent is an odd function because sine is odd function it changes the sine of the function if you change the sine of the argument right so sine of minus x equals minus sine of x now cosine is an even function which means it's the same for minus x and x if you want to check it again draw a picture of the unit circle which means that if you divide one over another wherever it's possible of course you will see that the tangent of minus x is equal to minus tangent of x which means tangent is an odd function it changes the sine of the function if the sine of the function if the sine of the argument is changing now to derive the property I'm talking about right now let's do it this way I change the sine of the argument which means I have to change the sine of the function now pi is a period so whether we add pi or subtract pi it will be the same as tangent x so this is another formula tangent of pi minus x is equal to minus tangent of x tangent of pi plus x or x plus pi is the same as tangent without changing the sine this pi is a period what else remains to be so asymptotes are at pi over 2 asymptotes pi over 2 plus pi is any natural number positive or any indigent number it can be negative as well so pi over 2 that's where your asymptotes and then 3pi over 2 minus pi over 2 minus 3pi over 2 etc and zeros are at 0 and pi etc at pi times n where n is any so these are the properties this is the function this is the graph what else regular manipulations with graph if you are like stretching, squeezing, shifting whatever remains exactly the same I don't want to go into this basically that's it about tangent again let's just remember that unit circle in classical sense is used to define sine and cosine and tangent is usually defined as a ratio obviously we can define it as ratio not of a sine over cosine but a ratio of originate to absolute of the point on the unit circle but traditionally this is kind of more appropriate I guess well that's it thank you very much and good luck