Graphs of Tangent and Cotangent Functions - Video Tutorials & Practice Problems

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1

concept

Introduction to Tangent Graph

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Hey, everyone. So in the last few videos, we've been kind of on this journey of looking at the various graphs of trigonometric functions. We've already seen what the sign and cosine look like as well as what the sin and cos look like. Well, in this video, we're now going to take a look at the graph of the tangent function at this sounds intimidating. Don't sweat it because we're going to learn in this video that just like the Kent and C function, the tangent function will also build off of the knowledge that we've already learned about these graphs like the sine and cosine for example. So without further ado, let's take a look at what some of these problems might look like in this course where we have to deal with the tangent function as a graph. Now, I want you to recall this identity which we learned about early on that says that the tangent of some value is equal to the sign over the cosine. So what we can do is use this to graph what the tangent looks like by taking the S and dividing it by the cosine values. Now, I'm going to go ahead and start with zero. And I know that I'm doing this a bit out of order. I'm not starting from the left side this time, but you're going to see why. So if I start with zero, notice that we're going to have at an X value of zero, that our tangent is S over cosine. Well, that's just gonna be zero divided by one which is zero. So that means we're going to start right about there on our graph. Now I'm going to do from here is go to the right and the left on this graph. So we start by going to the right at an X value of pi over four, this is going to give you square root 2/2 divided by square root 2/2, which is just a value of one. So pi over four, we have an output of one for our tangent graph. Now, if I go to the left, we're going to have negative square root of 2/2 divided by that same value but positive and doing this will give you a value of negative one, meaning we're going to be down here on our graph. Now, what happens if I go to pi over two and negative pi over two on the X axis? Well, I want you to notice something in both these situations. What we end up having is this case where we have to divide by zero, we're going to have negative one over zero and one over zero, we learn in math that you cannot divide by zero. So that means that these values are undefined. And because we get undefined values at negative pi over two and at pi over two, that means we're going to have asymptotes show up. So we're going to have an Asymptote right about here and an Asymptote right about there and call that an Asymptote just means the function is approaching infinity at that point. So that means our graph is going to look something like this. We're going to go down and then to the left and then we're going to go up into the right. So this would be the curve for our tangent function. Now, there's a couple of things you should know about the tangent graph. The tangent graph actually has some similarities to the second graph which we saw in the previous video. And one of the similarities is that the asymptotes are at the same values on the X axis. Notice how we have asymptotes at negative three power two negative pi over two pi over two and three pi over two. Well, for our tangent graph, we can see these asymptotes are the same. And it turns out that for the tangent graph, all of the asymptotes or places where the curve repeats are going to be when the cosine of X is equal to zero, which are going to be at odd multiples of pi over two. So because of this, on our tangent graph, we're going to see another Asymptote and negative three pi over two and an Asymptote at positive three pi over two. Now, the way that the tangent graph is different than the seeking graph is, well, obviously this curve is different. But it turns out that for the tangent graph, the tangent graph actually has a period of only pi right, for the second graph, we can see that what happens is we get this smiley face, then this frowny face and then the curve is going to repeat. So we'll get another smiley then another frowny. So it repeats after two pi units because the distance from here to there on the X axis would be two pi, well, it turns out that the tangent graph will actually repeat every pi units. So rather than getting some kind of different curve, you're going to get this same curve right about here and you're going to get the same curve right about there. So every time you travel a distance of pi on the X axis, you're going to get your curve repeating. And what this does is actually changes the period for the tangent. So in the past, we've seen that the period for the sine cosine co and see is two pi over B. Well, it turns out when dealing with the tangent, the period is only pi over B. So that's something you have to watch out for when dealing with the tangent graph. But the nice thing about tangent graph is that all of the same transformation rules apply that have been used for sine and cosine aside from the period change now to make sure that we're understanding this, let's try an example where we have to graph a function which deals with the tangent. So here we're asked to graph the function Y is equal to the tangent of pi over two times X. And the way that I'm going to do this is first figuring out what the period of our graph is going to be. Well, we're called that the period for tangent as we just learned is pi over B rather than two pi over B. And the B value is what we have in front of the X. So we're going to have pi divided by pi over two. Now, what I can do is take this fraction we have in the denominator and flip it and bring it to the numerator. So I can flip this right here and we're going to get two pi divided by pi because this will stay in the denominator. And then those P are going to cancel, giving us a period of two. So if our period is two, that means that every two units we're going to have our curve repeat. So what I'm going to do is label the X axis like this. We're going to say that we have points here 123 on the X axis. And then we have negative one, negative two, negative three. Now we learned that how the tangent graph starts right here at the middle and then it curves up into the right and down into the left. And we also know that this tangent graph is gonna have a period of two. So what I can do is draw asymptotes here at negative one and I can draw another Asymptote at positive one because this would be a period of two on the X axis. So what I can do is draw a curve that looks something like this. So this would be the tangent graph now because the period is two. If I start here where this asym tote is and I go two units to the right, we're going to have another Asymptote at positive three. And if I go to negative one, we can go two units to the left and get another Asymptote at negative three. And this curve is going to repeat between each of these asymptotes. So we're going to get the same curve right about there and the same curve back here. So this is what the graph would look like and that is the solution to the problem. So hopefully, this gives you a better understanding of how to work with graphs when you're dealing with the tangent function. Thanks for watching and please let me know if you have any questions

2

Problem

Problem

Below is a graph of the function $y=\tan\left(bx\right)$. Determine the value of b.

A

$b=\frac14$

B

$b=\pi$

C

$b=2$

D

$b=\frac12$

3

example

Example 1

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3m

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Hey, everyone. So here we have an example where we are asked to graph the function Y is equal to one half times the tangent of X minus pi over two. Now, to solve this problem, what I'm going to do is I'm first going to graph the function Y is equal to one half times the tangent of X. So I'm going to ignore this pi over two for now, but we will get to this later. So let's start with this function to find Y is equal to one half the tangent of X. Well, let's just recall what the tangent graph looks like. I know for the tangent graph, we have asymptotes at negative pi over two and at pi over two. Now we do have more asymptotes on this graph. We're going to have other asymptotes at odd multiples of pi over two. But for now, I'm just going to draw these two. Now as for the curve, I know the curve starts here at the middle and then it goes up into the right and then down and to the left. Now, another thing that I recall for this tangent is that we have points at one or at pi over four comma one and then at negative pi over four comma negative one. So the graph ends up looking something like this, but notice that we have a one half in front of our tangent and this one half as you may recall, changes the amplitude or tallness of our graph. So rather than having these points at one and negative one on the Y axis, these are going to be reduced. So we're then going to have a point instead at one half and then at negative one half for these two values of power four and negative power four respectively. So that means the graph is going to look a little bit more like this where we can see that we're a little bit shorter than we were before. So this right here would be the curve for Y equals one half the tangent of X. Now, our last step for solving this problem is going to be to incorporate this pi over two. So we need to figure out what Y equals one half times the tangent of X minus pi over two looks like. Well, something that I can see is that we have a pi over two here, which is going to shift our graph in some sort of way and define the shift. Well, recall that we can find the H over B ratio which tells us how much we've shifted to the left or to the right now see that we have a minus sign here, which means that this is actually a positive H value. So we have positive pi over two divided by B which in this case is any number in front of the X. But since there's nothing there, we can just write one, so we're going to have H over B is equal to just pi over two because this one of the denominator is just going to keep this the same. So that means that our graph is going to be shifted pi over two units to the right, because we got a positive H over B. So what I can do is take this point right here and I can shift it pi over two units to the right. In fact, I can take this Asymptote and shift it pi over two units to the right, which would put us at pi, then I can take this Asymptote and shift it pi over two units to the right, which would put us at zero. So this is what the graph is going to look like. And because we have asymptotes at pi and then at zero, we're going to have another Asymptote at negative pi on the X axis. And what I can do for here is continue drawing these curves since they're going to repeat every pi amount of units. So I'm going to have another curve which is right back here. It's going to go look like this and then like that, it's going to be the same as the curve that we had before because this curve just continues repeating. So the curve is going to be back here as well. It's going to go up and then down and then this curve is going to repeat over here, going up and down to the left as well. So this is what our curves are going to look like. This is the graph for the tangent and that is the solution to this problem. So I hope you found this video helpful. Thanks for watching.

4

concept

Introduction to Cotangent Graph

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5m

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Welcome back everyone. So at this point, we should be familiar with the graphs. For most of the trigonometric functions, we've talked about the sine and cosine graphs. The see and coin we even looked at the tangent graph in the previous video. Now, last but not least we're going to be taking a look at the cotangent graph. And this might sound a bit complicated just because learning about another trigonometric graph might feel kind of exhausting at this point. But what we're going to learn is that the co tangent graph is actually closely related to the tangent graph. And if you recognize how these two are related, then sketching this graph is actually pretty straightforward. So without further ado let's get right into things. Now recall how based on reciprocal identities, the code tangent is just one over the tangent. So we can use this fact to draw the graph for the code tangent. Now, the first thing that I'm gonna look for is where we can draw these asymptotes because this will tell us how wide each of the cycles of our graph is. Now, if I look at the tangent graph, I can see that we reach zero at negative pi zero and pi so we reach zero on the y axis at all these points. And it is a fundamental math rule that you cannot divide by zero. So any place that we see zero on the tangent is going to be a value that cannot exist on our code tangent graph. So that's going to be where we draw the asymptotes. So there's going to be an Asymptote right here at negative pi there's also going to be Asymptote here at zero on the X axis. And there's going to be an Asymptote over here at positive pi. Now, from here, we need to draw the curves and it turns out that the curves for the cotangent graph are similar to the tangent graph, but they're going to be flipped upside down. So rather than having curves that go from bottom left to top, right, we're going to have curves that go from top left to bottom, right. So the graph is going to look something like this. Now, I want you to notice all the points on the cotangent graph where we touch the X axis. This is where our output is going to be equal to zero. And we can actually see that these values make sense when we look at the tangent graph because notice that we touch the X axis at negative three pi over two. And on our tangent graph, our graph gets really, really small when we get to negative three pi over two in the negative direction. So basically we go to negative infinity. And whenever you divide a number that is approaching infinity or whether it's a negative or positive number, then your entire fraction or output is going to be equal to zero approximately or it's going to approach zero. So that is why we have zero at negative three pi over two at negative pi over two at pi over two. And at three pi over two, you can see at all of these points on our tangent graph, we're either getting really, really big there towards infinity or really, really small towards negative infinity. Now, one of the main similarities between the tangent and cotangent graph is the distance on our x axis which we have the curves repeat. Because notice for our tangent graph that we repeat from negative pi over two to pi over two, which is just a distance of Pi and it is a distance of pi for each of these curves on this graph. So that is one of the nice things about the tangent graph is we repeat every pi units. And it turns out that the cotangent graph is the exact same thing because if you look, we can see here our graph goes negative pi to zero. And that's just the distance of PI and we also go from zero to pi which is also a distance of pi. So it turns out that this graph repeats every pi amount of units as well. Now, because these graphs have the same repeating behavior, it turns out their periods are also going to be the same. So just like the period of the tangent was pi over B, the period of the co tangent is also pi over B. Now, the way these graphs are different from each other is the values of their asymptotes. So for example, if we take a look at the T graph, we can see that these asymptotes will always show up at odd multiples of pi over two. So we have an Asymptote at negative three pi over two negative pi over two, we have an asto pi over 23 pi over two. Then if we kept extending this graph, we'd also have asymptotes at five pi over two and seven pi over two. Now notice for our Cohan in graph, our asymptotes are at negative pi, they're at zero, they're at pi if we kept extending this graph, we would see asymptotes at two pi three pi four pi. So it turns out for the tang graph, the asymptotes are always going to be at integer multiples of pi. So the location of the asymptotes on the X axis as well as the fact that these curves are flipped upside down are really the two main differences between the code tangent and tangent graph. Now, to make sure we understand this concept. Well, let's try an example where we have to deal with the code tangent. Now, to solve this problem, what I'm first going to do is figure out where the asymptotes are going to be located on this graph. And I can do that by looking at what our code tangent is. Now, we're called that the period of the code tangent is just pi over B. Now, what I can see here is that our B value is the value we have in front of the X. And I can see that B value is equal to pi. So we're going to have pi over pi and whenever you divide one number by itself, it's just going to be equal to one because they're both going to cancel and give you one. So that means that our period is equal to one. So what I can do on our graph is I can draw points every one unit on our X axis. I'll go 123 to the right and they'll go negative one, negative two, negative three to the left. Now, the period of this graph is one, meaning I can draw these asymptotes for every one unit. So we're going to have an Asymptote here at zero, we have it at one and then at two and then at three and then these asymptotes are also going to repeat behind the graph as well. So we have an Asymptote at negative one, negative two, negative three. And this is what the asymptotes are going to look like. Now, my last step for solving this problem is going to be to draw the curves and we called it curves go from top left to bottom right between these asymptotes. So what I'm going to do is have a curve that looks something like that. And then these curves are just going to keep repeating for each period of this graph because that's what the repeating function is going to do. So this is what the code tangent is going to look like. So this right here is the graph of our function and the solution to this problem. So that is how you can deal with graphs when it comes to the co tangent function. I hope you found this video helpful. Thanks for watching and please let me know if you have any questions.

5

Problem

Problem

Below is a graph of the function $y=\cot\left(bx+\frac{\pi}{2}\right)$. Determine the value of b.

A

$b=\frac14$

B

$b=1$

C

$b=2$

D

$b=\frac12$

6

example

Example 1

Video duration:

3m

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Hey, everyone. So let's see how we can go about solving this example. So here we are asked to graph the function Y is equal to negative two times the cotangent of 1/4 X. Now, what I noticed in this problem is we have quite a few things going on. We have a negative sign here which is going to cause our graph to be flipped or reflected over the X axis. I also noticed we have this two here which is going to change the amplitude or tallness of our graph. And we have this 1/4 on the inside of our code tangent which is going to change the period. So where do we even get started with this? Well, what I like to do is start with what I'm familiar with and what I'm going to do is I'm going to ignore this amplitude and negative sign out front. I'm just going to focus on Y equals the cotangent of 1/4 X. And the reason that I'm focusing on this is because I know how to deal with the period of a cotangent. And I know what the general cotangent graph looks like. So I can build off of something. Now, the period for a cotangent graph is going to be pi over B where B is the number that is in front of the X. Since I can see that that's 1/4 I get that we have pi divided by 1/4 and you can flip this fraction and bring it to the top, which is going to give you four pi over one which is just four pi. So the period is four pi and something that I know about the co tangent graph is the co tangent graph has an Asymptote that starts at an X value of zero. So since we start in an X value of zero, and our period is four pi when we have this B value in front of X, that means that starting at zero, we're going to repeat every four pi units. So I can draw another Asymptote here at four pi and I can draw another Asymptote over there at eight pi. Now, something I know about the cotangent graph is that it goes up into the left and then goes down into the right and it reaches zero on the X axis right in between the two asymptotes in the middle. Now, what you should typically see with a cotangent graph is you're going to see points that are in between these two values and in between those two values. So this over here should be pi and this should be three pi and that makes sense because we'd have pi two pi three pi four pi and at these values, you should see an output of one. So what we should have is an output right about there, which would be at one, we should see an output right here, which would typically be at negative one. So adjusting our curve a little bit, this is what the function will look like for the co T of 1/4 X. And then you would get this curve continuously repeating as we go along. But this is not what we're trying to find. We're not trying to find the code tangent of 1/4 X. We're trying to find negative two times the code tangent of 1/4 X. Now this number out in front here, the two, this is going to change the amplitude or basically vertically stretch our graph. So rather than having an output of one, when you're at pi, you should actually have an output of two at this value. And then rather than having an output at negative one for three pi you should actually have an output at negative two. So adjusting our curve, the graph should look a bit more stretched vertically when we incorporate this too. But there's one more thing we need to change about our graph and that's the negative sign. This negative sign is going to take our whole graph and it's going to flip it over the X axis. So doing this, we actually need to in essence reverse these two points on the Y axis. So we should have a point right here at pi negative two and the point right here at three pi positive two. So for the graph Y equals negative two times the cotangent of 1/4 X, it should look something like this where we start at the lower left. And then we go ahead and go up reach a zero on the Y axis at two pi and then we come up here and go to the top right. So this is what our curve should look like. And because the Kana is a repeating graph, as we've learned in previous videos, this curve should continue happening between each of these asymptotes. So this would be the graph of our function and the solution to the problem. So I hope you found this video helpful. Thanks for watching.