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મુખ્ય વિષયવસ્તુ

વર્તમાન સમય:0:00કુલ સમયગાળો :10:12

so what I hope to do in this video is get some familiarity with the graph of tangent tangent of theta and to do that I'll set up a little unit circle so that we can visualize what the tangent of various Thetas are so let's say that's a y-axis this is my x-axis that is my x-axis and the unit circle would look something something like this and we already know this is all of a refresher of the unit circle definition of trig functions that if I have an angle an angle theta where one side is the positive x-axis and then the other side so this is the other side so the angle is formed like this that where this ray intersects the unit circle the coordinates of that the x and y coordinates are the sine of theta sorry the x coordinate is a cosine of theta so it's the cosine of theta comma sine of theta so the x coordinate here is cosine theta the y coordinate there is the sine of theta but we're concerned about tangent of theta well we know the tangent of theta is the same thing as the sine of theta over the cosine of theta or if you're starting from the origin and you're going and you're you you're taking the value of essentially the y coordinate the y coordinate over the x coordinate it's essentially the slope of this line it's going to be this is going to be your change in Y change in Y over change in X this right over here is going to be the slope the slope I guess you could say of this ray right over here and so that's going to help us visualize what the tangents of different Thetas are so let me clean up my unit circle a little bit just so that we can all right there we go so now let's make a let's make a table so let's make a table so if so for various Thetas let's think about what tangent of theta is going to be tangent of theta so maybe the easiest one if theta is a zero radians so if it's zero radians what is the slope of this ray well that raises the slope is zero as X changes why does it change at all now let's think about out and I'm just going to pick values that is very easy for us to think about what the tangent of those values are and that will help us form they'll help us think about the shape of the graph of y is equal to tangent of theta so let's take let's take PI over PI over 4 radians so this one right over here theta is equal to PI over 4 now why is that interesting and sometimes it's easier to think in degrees it's a 45-degree angle this here your x-coordinate and your y-coordinate is the same you might remember it square root of 2 over 2 square root of 2 over 2 but the important thing is whatever you move in the X direction you move the same in the Y direction so the slope of this ray right over here is going to be equal to 1 or another way of thinking about it tangent of theta is going to be equal to 1 or sine of theta over cosine of theta the same thing so you're going to get 1 so if you put so let me just clean that up here just because I'm going to keep reusing the same unit circle so if I have theta is PI over 4 then the tangent of theta is going to be equal to 1 now what if theta is equal to negative PI over 4 so that is this right over here so with X so let me just draw a little triangle here so when X this x-coordinate over here is square root of 2 over 2 we know that we've seen that multiple times square root of 2 actually let me label a little let me label it a little bit better so here our theta is equal to negative PI over 4 radians now or you could if you like to think in degrees this would be negative 45 degrees and now your your sine and cosine of this angle are going to be the opposites of each other the cosine is square root of 2 over 2 the x-coordinate of where this intersects is square root of 2 over 2 the y-coordinate here is negative square root of 2 negative square root of 2 over 2 so what's the tangent well it's going to be your sign over your cosine which is going to be negative 1 and you see that for however much you move in the X direction you move the opposite of that you move the negative of that in the y-direction and so let me clean this up a little bit because I want to keep reusing my unit circle so there you go and so this is going to be negative one this is going to be this is going to be negative one and so actually let's just start plotting a few of these points so if we assume that this is the theta axis if you can see that that's the theta axis and if this is the y axis that's the y axis we immediately see tangent of zero is zero tangent of PI over four is one with the king in radians tangent of negative PI over four is negative one now let's think now if you just saw that you might say well maybe this is some type of a line but we'll see very clearly it's not a line because what happens what happens as our tent as our angle gets closer and closer to as our angle gets closer and closer to PI over two what happens to the slope of this line so that is theta we're getting closer and closer to PI over two well this ray I guess I should say is getting closer and closer to approaching the vertical so it's slope is getting more and more and more positive and if you go all the way to PI over two the slope at that point is really undefined but it's approaching one way to think about it is it is approaching infinity so as you get closer and closer to PI over two so I'm going to make a I'm going to draw essentially a vertical asymptote right over here at PI over two because it's not going to be I guess one way you can think about it's approaching infinity there so this is going to be looking something like this it's going to be looking something like this the slope of the Rays you get closer and closer to PI over two is getting closer and closer to infinity and what happens when what happens when the angle is getting closer and closer to negative PI over two is getting closer and closer negative PI over two well then the slope is getting more and more and more negative it's really approaching negative it's approaching negative infinity so let me draw that so once again not quite defined right over there we have a vertical asymptote and we are approaching we are approaching negative infinity we are approaching negative infinity so that's what the graph of tangent of theta looks just at just over this section of I guess we could say the theta axis but then we could keep going there we could keep going because if our angle right after we cross PI over 2 so where let's say we've just crossed PI over 2 so we went right across it now what is the slope what is the slope of this thing well the slope of this thing is hugely negative right it looks almost like what I just drew down here it's hugely negative so then the graph jumps back down here and it's hugely negative again it's usually negative and then as we as we increase our theta as we increase our theta becomes less and less and less negative all the way to when we go to what is this all the way until we go to let me plot this this angle right over here now what is this angle this if I haven't told you yet this let's say that this angle right over here is 3 PI over 4 now why did I pick 3 PI over 4 because that is PI over 2 that is PI over 2 plus PI over 4 or you could say 2 PI over 4 plus another PI over 4 is 3 PI over 4 and the reason why this is interesting is because it is another it's forming another I guess you could say PI over 4 PI over 4 PI over 2 triangle or 45-45-90 triangle where the x and y coordinates or the x and y distances have the same magnitude but now the X is going to be negative and the Y is positive so the slope here is going to be the slope at the same slope as we had for negative PI over 4 radians we're going to have a slope of negative 1 so at 3 PI over 4 we have a slope of negative 1 then we increase our angle all the way to PI now our slope is back to 0 our slope is back to 0 and then as we go beyond that as we go to as we increase by another as we increase by another PI over four our slope goes back to being positive one our slope goes back to being positive one and then once again as we approach 3 PI over 2 our slope is becoming more and more and more positive getting approaching positive approaching positive infinity this slope notice if you move a little bit in the X direction you're moving a lot up in the Y direction so once again so now the graph is going to look like this let me do it in a color that you can actually see the graph is going to look something something like something like this and it will just continue to do this it will just continue to do this every every pi radians every which maybe we do that as a dotted line every PI radians over and over and over again so let me go back PI I can draw these asymptotes I can draw these asymptotes and so let me draw that and that and so the graph of tangent the graph of tangent of theta is going to look is going to look something something like this and we could obviously it's periodic we could just keep doing it on and on and on in both directions