Everyone welcome to our lecture on other periodic functions. In this lecture will talk about tangent, cotangent, secant and cosecant. The most basic trig functions, sine and cosine, we've defined these already. And we just want to sort of manipulate these functions and involve quotients and reciprocals of these. So if you're getting comfortable with sine and cosine that your foundation, and from that we're going to define the tangent, cotangent, secant and cosecant. Here we go, let's begin. So first off, let's define tangent of x as a function. Remember this is tangent of x, this is not multiplication, but this will be sine of x divided by cosine of x. So this is a quotient of the two functions that we know and then we love. We will have secant of x, so this is not sec, as S-E-C is abbreviated, but its secant of x. This is the reciprocal which means is 1 over, so 1 over cosine of x. Of course we have cotangent, we review that cot, cotangent of x, that's a cot. It's cotangent of x this is 1 over tangent, so the reciprocal of tangent. You could write as 1 over tangent, but oftentimes want to describe things in terms of sine and cosine this is where if we get really good at sine and cosine, we don't need to learn everything else. So sine and cosine, flip that and you get cosine over sine. And last but not least, cosecant, this is abbreviated as csc of x. This is the reciprocal of sine, is the reciprocal of sine of x, okay? Now one little thing that's usually tricky, some people get confused, the reciprocal of sine is cosecant, it starts with a C. And the reciprocal of cosine starts with an S, so of course they don't match, right? That would make life easy, but they don't, okay? A couple things with all fractions, you gotta watch out for just little things. We have fractions for tangent, and so when is this defined? What's the domain we could ask for these functions while the domain for this? Domain for tangent, cosine x is whatever cosine x is not 0. So all these functions with cosine on the bottom, they are defined whenever x is not 3 pi over 2, or just some multiple of pi plus pi and a half. So watch out for fractions, as always, these are just still number divided by number, the denominator cannot be 0. So tangent secant functions, they have fractions, and then for the fractions that have sine in their denominator, cotangent and cosecant in particular, they are not defined when sine of x is 0 or when x is multiples of pi, okay? So put these down somewhere, keep these definitions handy and we'll go over these and then we'll do some examples to get more familiar with them. If we start introducing these functions, let's draw the unit circle and let's label our quadrants here. So we have our nice unit circle, we have quadrant 1, we have quadrant 2, quadrant 3, and quadrant 4, of course going counterclockwise. Where are these things positive, where are these things negative? So remember that x is the cosine of the angle formed off the x axis and y of course is the sine. So cosine and sine in particular are going to be positive in quadrant 1. So over here we have x is positive and y is quadrant 1. So cosine theta is positive and sine of theta is also positive, so greater than 0 of course means positive. And I guess it could be 0 again, but in quadrant 1 is the idea. So now if I start looking at tangent, tangent is the quotient of sine over cosine. So positive number divided by a positive number is also positive. So sine, cosine, tangent, they're all positive quadrant 1. If I start taking their reciprocal functions, so the reciprocal of cosine 1 over is secant, so secant theta, 1 over a positive number. Think about that for a second, that's positive, cosecant theta, that's also positive, 1 over positive and then cotangent is also positive. So normally there are six trig functions that we talk about, and we're going to list them in sort of table rows of free cosine, sine, tangent, secant, cosecant, cotangent. They're all positive. So what I'd like you to do now is sort of pause the video and pick a random quadrant and think about what is the sine of all six of these functions, what is the sine? We'll do one more here and leave the rest as exercises. But let's pick quadrant 3 for no good reason. So quadrant three we-re in the negative x-axis, we're also in the negative y-axis as well. So cosine of theta which is your x value, is negative, sine of theta, also negative. And now. Tangent, be careful. Tangent theta is negative divided by negative. Two negatives make a positive. So tangent, said the sinus which era positive. Then take reciprocals, reciprocals does not change the sign. If I'm negative two, when I flip it on negative 1/2. So secant of theta would also be negative. Cosecant of theta is also negative and of course cotangent would be positive. Okay? So watch out for these signs as you work your way through the quadrants and whenever you're trying to do evaluations, it's good to keep in mind. Ask yourself first what value should I get? Should I get a positive number? Should I get a negative number? Zero, who knows? But keep this in mind as you go through quadrants. Let's have some example where we're going to evaluate all six trig functions for a given angle. So let's start out with a nice angle in quadrant one. So let's say, Theta is π over 4. Of course this is in radians. And here's what I want. I want sine of this angle. I want cosine of the angle, I want tangent of the angle. So there's three right there. And then I want the reciprocals. Cosecant of the angle, secant of the angle one over sin, one over cosine, and then of course cotangent. Now this is where if you stare at this and sort of pause for a minute, it's, My gosh, this is terrifying. This is crazy. I have 6 things to solve. What I'm trying to say is if we get really good at cosine and sine, you only have two things to figure out, what is sin of π over 4? What is cosine π over 4? This you might need to look up. Remind yourself, what is of course were in quadrant one with power before. This is about 45 degrees. So Theta is π over 4. This is right where 45 degrees half a square. You get nice things that are equal here, somewhere some on your notes or in past videos, we've done that. Sine and cosine, their equal root 2 over 2. So post a video and look that up and remind yourself what that is, if you need to. Now you may be staring at this mountain. I have four more to go. You don't, right? Why? Tangent is sine over cosine, so we're going to use the prior knowledge. Say sine over cosine. That's two numbers or two over two or two over two. That's when the numerator is equal to the denominator. That of course, is just one. Cosecant, how am I ever going to find that? Wait a minute. It's one device. The reciprocal. I just flip it is like no math here. Just flip it, two over root two, yeah, leave that root two downstairs, don't rationalize the denominator. You'd be a rebel, Cosecant flip cosine. Well it's the same thing. Two divided by root two. Cotangent, flip one. I could do that, one, one over one is one. So you get all these nice things over here. Once in awhile just because I know there's some people yelling at me out there in Internet land. They say you can have square roots in the denominator. You can, I promise you can if you don't believe me, go plug this in your calculator. I assure you it will work. If you do want to rationalize the nominator, this is just bugging you to see that and you want to multiply by root two of root two. You can certainly do so, in your numerator, you get two root two and then you get root two times root two, which of course is two. The two cancel and this is a cleaner cleaned up version. It's just square root of two. If you want to write it a square root of two, be my guest. Either way is fine in all honesty and most textbooks they will rationalize the nominator. So it's good to get comfortable seeing it both ways. Let's do one more. Let's find all six trig functions for another value of theta. Let's do 150 degrees, we're going to work in degrees right now, so 150 degrees. Where are we on the unit circle? We're not in quadrant one anymore. We've gone to what quadrant? Quadrant two. Quadrant two, so it's 180 and then back up 30. So rotate around the circle, 180 and then back up 30. There's your 150 degree angle. Nice obtuse angle and this little baby reference angle is 30 degrees. So we have 150 and then it's reference angle is 30 degrees, so in particular, we have a reference angle and I want to find, let's do one at a time here. Let's just get sine and cosine to start. Hopefully we're good at this, sine of course is the Y value and the Y value of the point on the unit circle is the same as the Y value of its reference angle. We turn any angle not in quadrant one into something in quadrant one. So we look at 30 degrees and it's positive in quadrant two, so this becomes sine of 30. So sine of 30 degrees is another way to look at this, and that of course we have, we've done this one before, this is 1/2. So remind yourself why that's 1/2 if you need to. Cosine of 150, that is of course all the same as cosine of 30 but, but, but, the X value for 30 were quadrant two, X is negative in quadrant two. If I start looking over in quadrant one, I'm going to turn the X value positive. So of course I need to. Really negate whatever answer I get for cosine of 30, cosine of 30 is root 3 over 2, so my final answer is negative root 3 over 2. If you got those two, give yourself a little pat on the back. That is really great. Once we have two, the rest is then smooth sailing, tangent of 150 degrees is sine over cosine, so this is I'll write it out so you see it. Sine of 150 degrees over cosine of 150 degrees and that of course is one half. Divided by negative root 3 over 2 hey look at that fraction divided by fraction. We know how to handle that. Keep the first one, keep change, flip. Keep the numerator one half change division to multiplication, and then flip the denominator. Don't lose this sine on that process, so we have negative two over root 3 cancel the twos we're left with negative one over root 3. But you got the rational, I know, I know. Just lazy, I want to do it, but I'll do it for you guys. Because I care minus 3 over 3, there we go so finally once you rationalize the denominator is negative root 3 divided by 3. Again, here the key thing is to be negative. Watch out for that the rest of them I think follow pretty, pretty straightforward cosecant of theta. Remember this is the reciprocal of sine. This is the reciprocal of one half flip it you get 2. Secant of Theta, this is the reciprocal of cosine. Flip it you get minus two over root 3. I'll let you rationalize the nominator if you want. If you do that, you get negative two root three over 3. And then of course cotangent of Theta is the reciprocal of tangent and that is negative three over root 3. So because we're in quadrant two, we have to watch out for not everything being positive. So just be very careful as you work this out with your signs. Oftentimes we will be using a scientific calculator or an online calculator to compute these values. And I just want you to call watch out for something here. There are no values most of the time for the reciprocal functions. So this is my warning. On a calculator, you will see the buttons like usually you'll see like sine of X. And then on top you'll see sine inverse for like if you hit second and the command. This is my warning to you. Be careful we haven't seen these yet, but we will. This is not one over sine. Which is cosecant. This is not so these buttons on the top that are called inverse functions we'll get to them in a future video, but just be aware they are not the reciprocal. If you want the reciprocal, you must actually type one over sine or one over cosine, or one over tangent. So just be very careful with that. Let's just practice a couple of calculator exercises, so grab your favorite calculating device and let's go through some of them. And here's what I want you to compute. Compute for me, secant of pi over 12, compute for me cosecant of 123 degrees and then compute for me cotangent of negative 12.4 degrees as well. And you can round these if you want to a couple decimals. Okay so just pause the video, practice these, but it's important to get these right and one of the things you have to watch out for is your calculator in radians. Is it in degrees? Can you go back and forth accordingly? Pi over 12 of course that is in radians. Secant, there's no button for secant on your calculator, so you would literally have to push one over cosine of pi over 12, so make sure you're using pi. Do not put 3.14. That introduces a rounding error a little too early on things. And if you do this, if you do this right, if you go to four decimals you get 1.0353 cosecant of 123 degrees now switch your calculator back two degrees. In that case there's no button for cosecant. You have to do 1 divided by sine of 123 degrees, right? So check this when you work this out you get 1.1924 and last but not least cotangent. Again, there's no button for cotangent. You have to type 1 divided by tangent and you can put it back into well you can leave in degrees I guess for this one minus 12.4. That's going to be, let's see when I type that in, 5.9551. Watch your rounding on your last decimal follow any directions about how to round these things, find these things too. And just be careful with the rounding. I've seen students get it right but they put the last number in wrong, and that breaks my heart. As with all functions we're going to want to know some graphs. These things we'll focus mostly on the graph of tangent as it comes up the most often, but you should know how to graph the other ones. Just the general shape of them or have to throw it inside of a graphing calculator. If you ever need to do it so the tangent function, this is a fraction, right? It's sine over cosine, so it's 0. Like water, it's when is it output zero, when the numerator zero, when sine function is zero and of course is undefined when the denominator is zero and cosine is zero. So this is going to be a graph that has some discontinuities to it. So tangent of X is sine of X over cosine of X. We have to watch out for this new fraction form. And when is it going to be positive? We can think of being a circle, we're going to be positive if the angle's in quadrant one. Since sine and cosine are both positive, we're going to be negative in quadrant two. Since sine is positive in cosine, so we're going to jump back and forth between positive and negative as we go out and do this thing. So let's start grabbing a couple values here. First, let's get some asymptotes here, this graph is not defined, so its domain is when- or has asymptotes, vertical asymptotes, VA. Abbreviate that for short when cosine of X is 0, so from there where is cosine X equals 0. Well cosine is the X value on the unit circle, so we're going to have cosine X at Πover 2. And so like 90 degrees at the very top of the inner circle, and then of course also at- draw this out a little bit, 3 π over 2 is at the very bottom of the unit circle. And you can look at the graphical side to see that, that forces the denominator B0, and that's going to mean that tangent is undefined. So we're going to have these asymptotes and then by symmetry also, we have negative π over 2 and go back one more. I could keep going, they go on forever, but we'll just draw a little segment of the graph here and we'll notice a pattern. So I'm going to draw these asymptotes with dashed lines, great. And now I'll try to draw the graph, pick a different color, maybe purple. So, what else do we know? So at zero, let's find some basic values here, what's tangent of 0? Well, that's sine of 0 over cosine of 0. Sine of 0, or 0 over 1 which of course is just 0. So we go right through the origin that describes our- a nice Y intercept and then we said we're positive from 0 to π over 2. So we start to compute some other values as you get close to this intercept, an asymptote of pi over 2, it approaches but does not touch as it gets more and more positive. On the back end, we said we were negative from 0 to negative π over 2. So we're going to get close but not touch our asymptote of negative π over 2, and you'll notice that this pattern continues. We have another nice intercept right at 2π where sine is 0, so at 2π or at pi I should do first or zero again, and the pattern will continue. So it has a sort of same shape and sort of structure as X cubed almost, but the difference with X cubed is one of these pieces is the entire graph of X cubed. Tangent is a periodic function, it repeats in a very nice symmetric pattern. These intervals are all supposed to be the same length. I didn't do a great job, but it goes on and on forever and ever and with all periodic functions, one of the questions you might want to ask yourself is, what is this period, how often? How long does it take for it to repeat itself, period inside of cosine is one lap around the circle 2π. For tangent, a period is just pi due to the quotient nature of tangent. So this is a nice graph, I know it's good to have the general shape of this graph. It does come up enough, not as much of course as sine and cosine, I'd say those two are the most that are used most often, but tangent's another good one to know. So now let's do an example where we're just going to manipulate tangent a little bit just to get more practice with it. So let's do an example. Let's graph y equals tangent of 2x. So I wanted you to understand how small changes to- there's a coefficient on x. What does it do to the graph of tangent? And so if we look at this thing and maybe you can play around with it and plot some points if you want to pause the video, what's going to happen is it's going to compress the graph horizontally, in particular by a factor of two. So if I draw the xy axis and I start plugging things in, I could ask myself, what is tangent of 0? And you can check once again that you go right through the origin of 0 and you want to start thinking about what is the relationship of sine of 2x and cosine of 2x? How does that change what this function was before? And one thing you might want to do is actually- where is the denominator zero? Now, if you want cosine of 2x to equal 0, well, that means that you have increased the frequency of cosine of x. So in particular, now you're going to have values like x equals π over 4 or 3π over 4. So you're shrinking these gaps. The period inside of tangent. So let's draw those here. So π over 4. 3π over 4, and by symmetry you can do it on the back end, negative π over 4. And negative 3π over 4. Drawing your dash lines to get your asymptotes, your vertical asymptotes and then draw the shape. There's no negative size. We're not squaring anything, so the general shape is Reserved the what this does is it, this is the two in it. It changes where the vertical asymptotes line. So then you draw your little disco functions in between. Make sure you pass right through the origin, maybe draw little more symmetric that I'm drawing it now. But this is the general idea of what the tangent looks like. The goal of these sketches with transformations is that you just have some idea of how these things work. You never need an exact size, if you really wanted to four decimals go grab a calculator and work it out. If you needed a drawing for a publication or for some article or whatever. Yeah, go to some more professional graphing site and grab a picture there. The idea of this is if someone's talking to you, if you're reading something, and you see tangent of 2x, you want to have some intuition without having to run calculator of what this means. And it's like talking to having a conversation and looking up in the dictionary every third word, you want to have some intuition of how these things work, so play around with it. Will see some next video where I do this and and you get to see how manipulations of the graphs occur. Real quick just the other graphs that I want you just have a general idea of are going to be y = cosecant x, we won't do any transformations of these, but just get the general idea. Cosecant x the reciprocal of 1 over sine. So of course is going to be related to 1 over sine of x, so why don't we first draw sine of x. It starts at the origin, goes up and then down, its height of course is one to negative one. So this min and that is max, there's a good old sine of x. And so what happens when you take a number and you flip it? So for example, when I take zero at the origins function open zero and flip it, I'm all of a sudden undefined. So everywhere sine of x is zero cosecant x will be undefined, can't take zero and throw it in the denominator. It's going to cause some asymptotes, some vertical asymptotes in particular. When the function reaches its max, sine of x reaches max of 1, one over one is just one. So wherever you have a peek or a valley, the cosecant function will also have, will match at this point. Then you have to realize just sort of the relationship between numbers in general. As numbers get smaller, their reciprocal gets bigger, think about that for a minute. So like if I take or vice versa, as numbers get larger, there reciprocal will get smaller, so maybe that's an easier way to think of it. If I go 2, 3, 4, 5, 6, 7, and then I say one half, one third, one fourth, one fifth, one sixth, one seven, I'm getting smaller. And with that property you can then fill in the general shape of this graph as sine gets smaller heads towards the origin, its graph gets bigger. So as it goes down, the graphical secant goes up and then vice versa. Once you realize that pattern for reciprocals, you can then draw the shape, so it's with any periodic function you'd like to know its period. So since it's just the reciprocal assign its period as well is to pi. Once you understand or see how cosecant behaves, then finding the graph of secant is very, very similar. This is one over cosine of x, so let's draw the parent function cosine of x first. Cosine of x is like horizontal shift of sine, starts at 0, 1 goes up, it goes down, maxes out at one as its minimum at negative one. And for all the same reasons when cosine is zero, then the reciprocal is undefined. So everywhere you have an intercept, x intercept or zero, you're going to have an asymptote for this function is no longer defined. Cannot take the reciprocal of zero, and then where the function is one, it's going to match, negative one is going to match. So now we start doing the same thing, the numbers get smaller, the reciprocals get larger and vice versa. And off it goes up, down and up, and we start to sort of build here the shape of procedure. So if you could imagine in your mind, now the leading the graph of cosine, the leading graph of sine from this picture, you get these use that sort of alternate with period 2 pi. Once again, so I'll write that down so period, that is 2 pi. So reciprocal's are very very similar to sine and cosine, people who struggle and they constantly are not sure about cosecant or secant, or cotangent or even tangent. It usually comes down to a sort of haven't quite grasped sine and cosine just yet. So you really want to make sure you have a good hand of sine and cosine. That means knowing the graph, knowing some values, knowing how to find values in different quadrants, knowing the sign. Everything comes down to sine and cosine, and you can always translate or convert questions about all the other trig functions, all the other periodic functions in the questions about sine and cosine. Okay, so take all this in mind, review sine and cosine as you need, and then we'll do some more problems in the next video, I'll see you then.
