Hi, everyone. Welcome to our lecture on other periodic functions. In this lecture, we're going to 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 manipulate these functions and involve quotients and reciprocals of these. So if you're getting comfortable with sine and cosine, that's your foundation. From that, we're going to now define the tangent, cotangent, secant, and cosecant. Here you 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 the quotient of two functions that we know and that we love. We will have secant of x, so this is not to sec as S-E-C as abbreviated but it's secant of x, this is the reciprocal, that means one over, so 1 over cosine of x. Of course, we have cotangent, we abbreviate that cot, cotangent of x, let's say cot, it's cotangent of x. This is 1 over tangent, so the reciprocal of tangent. You can write as 1 over tangent but oftentimes, we 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. Last but not least, cosecant, and this is abbreviated as csc of x, this is the reciprocal of sine of x. One little thing that's usually tricky, some will 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, that would make life easy, but they don't. A couple things with all fractions, you got to 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. Well, the domain for this, where is it? There it is. The domain for tangent cosine x is whatever cosine x is not zero. So all these functions with cosine on the bottom, they are defined whenever x is not like pi, 3 pi over 2, or just some multiple 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 zero. So tangent secant functions, they have fractions. Then for the fractions that have sine in their denominator, cotangent and cosecant in particular, they are not defined when sine of x is zero or when x is multiples of pi. 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 positive quadrant 1. So cosine is positive, cosine theta is positive, and sine of theta is also positive. So greater than zero of course means positive. I guess it could be zero again, but in quadrant 1, this 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 in quadrant 1. If I start taking their reciprocal functions, so the reciprocal of cosine, 1 over it 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 talked about, and we're going to list them in a table of rows of three: cosine, sine, tangent; secant, cosecant, cotangent, they're all positive. So what I'd like you to do now is 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 we'll leave the rest as exercises, but let's pick quadrant 3 for no good reason. So quadrant 3, 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. Now tangent be careful, tangent Theta is negative divided by negative. Two negatives make up positive. So tangent, they said, the sine wish were positive. Then take reciprocals. Reciprocals does not change the sign. If I'm negative two and I flip it, I'm negative a half. So secant of Theta would also be negative. Cosecant of Theta is also negative, and of course, cotangent would be positive. 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 get a negative number? So there's zero, who knows? But keep this in mind as you go through the quadrants. Let's do an example where we're going to evaluate all six trig functions for a given angle. Let's start out with a nice angle in Quadrant 1. So let's do Theta is Pi over four. Of course, this is in radians. 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. Then I want the reciprocals, cosecant of the angle, secant of the angle, one over sine, 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 like, oh my gosh, this is terrifying, this is crazy. I have six 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 sine of Pi over four? What is cosine of Pi over four? This you might need to look up, or remind yourself what it is. Of course, we're in Quadrant 1 with Pi over four, this is about 45 degrees, so Theta is Pi over four. This is right where, remember Pi over four, 45 degrees, half a square. You get nice things that are equal here. Somewhere on your notes, or in past videos, we've done that sine and cosine they're equal root two over two. So pause the video, and look that up to 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 to say sine over cosine, oh, that's two numbers, root two over two, root over two. That's like when the numerator is equal to the denominator. That of course is just one. Cosecant, how am I ever going to find that? Oh, wait a minute. It's a reciprocal. I just flip it. There's like no math here. Just flip it. Two over root two. Yeah, leave that root two downstairs. Don't rationalize the denominator. You be a rebel. It's cosecant. Flip cosine. Well, it's the same thing. Two divided by root two. Cotangent, flip one. I could do that. One over one is one. So you get all these nice things over here. Once in a while, just because I know there's some people yelling to me out there in Internet land. They say, "You can't have square roots in the denominator." You can. I promise you, 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 denominator, 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 the numerator, you get two root two, and then you get root two times root two, which of course is two. The two's cancel. This is a cleaner, cleaned-up version. It's just the square root of two. If you want to write it as square root of two, be my guest. Either way is fine. In all honesty though, most textbooks, they will rationalize the denominator, 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 1 anymore. We've gone to what quadrant? Quadrant 2. It's like 180, and then backup 30. So rotate around the circle 180, and then backup 30, there's your 150-degree angle. Nice obtuse angle, and its little baby reference angle is 30 degrees. We have a 150, and then its reference angle is 30 degrees. So in particular, when we have a reference angle, let's do one at a time here. Let's just get sine and cosine to start. Hopefully, we are good at this. Sine, of course, is the y-value. 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 1 into something in Quadrant 1. So we'll look at 30 degrees, and it's positive in Quadrant 2, so this becomes sine of 30. So sine of 30 degrees is another way to look at this. That of course we have, we've done this one before. This is one-half. So remind yourself why that's one-half, if you need to. Cosine of 150, that is of course all the same as cosine of 30. But the x value for 30, we're in quadrant 2. x is negative in quadrant 2. If I start looking over in quadrant 1, 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've got those two, give yourself a pat on the back, that is really great. Once we have 2, 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 the division to multiplication, and then flip the denominator, don't lose the sign on that process. So we have negative 2 over root 3, cancel the 2s, we're left with negative 1 over root 3, which are the rational, I know, this is lazy, I want to do it. But I'll do it for you guys because I care. Minus 3 over 3, here we go. So final answer. 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 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 2 over root 3. I'll let you rationalize denominator if you want. If you do that, you get negative 2 root 3 over 3. Then of course cotangent of Theta is the reciprocal of tangent, and that is negative 3 over root 3. So because we're in quadrant 2, we have to watch out for not everything being positive, so just be very careful as you work this out with your signs. Oftentimes, you will be using a scientific calculator or all my 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 the secant on the comment. This is my warning to you, be careful, we haven't seen these yet, but we will, this is not 1 over sine which is cosecant. This is not, so these buttons on the top 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 1 over sine or 1 over cosine or 1 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 one I want you to compute. Compute from a 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. You can round these if you want to a couple of decimals. So just pause the video, practice these, but it's important to get these right. One of the things you have to watch out for is you calculator is in radians, is in degrees? Can you go back and forth accordingly? Pi of 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 we go to four decimal, you get 1.0353. Cosecant of 123 degrees, now switch your calculator back to degrees. In that case, there's no button for cosecant, you have to do one divided by sine of 123 degrees. So check this. When you work this out, you get 1.1924. 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, 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. I'm going to find these things too, and just be careful with around it. 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. Remember how to throw it inside of a graph and calculator if you ever need to do it. So the tangent function, this is a fraction, the sine over cosine, so it's zero. When is it output zero? When the numerator is zero, when sine function is zero, and of course, it's undefined when the denominator is zero, when 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. When is it going to be positive? We can think of the inner circle. We're going to be positive if the angles in quadrant 1, since sine and cosine are both positive, we're going to be negative in quadrant 2 since sine is positive and 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 graph in a couple of values here. First, let's get some asymptotes here. This graph is not defined, so its domain is when or has vertical asymptotes, VA, abbreviate that for short, when cosine of x is zero. So from the reminder, where's cosine x equals zero? Well, cosine is the x value on the inner circle, so we're going to have cosine x at Pi over 2 is 90 degrees at the very top of the inner circle. Then of course, also at, let me try this out a little bit, 3Pi over 2 is at the very bottom of the inner circle. That is, and you can look at the graph of cosine and see that, that forces the denominator to be zero 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 Pi over 2 and we'll 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 the pattern. So I'm going to draw these asymptotes with dash lines, and now I'll try to draw the graph, pick a different color, maybe purple. So what else we know? So add zero for some basic values here, like what's tangent of zero? Well, that's sine of zero over cosine of zero. Sine of zero is 0 over 1, which of course is just zero. So we go right through the origin that describes our nice y-intercept, and then we said we're positive from zero to Pi over 2 and we're also going to show. So you start to compute some other values as you get close to this intercept and asymptote at Pi over 2. It approaches, but does not touch as it gets more and more positive. On the back-end, we said we're negative, from zero to negative Pi over 2. So we're going to get close, but not touch our asymptote at negative pi over 2. You'll notice that this pattern continues, we have another nice intercept right at 2Pi with sine of zero. So add 2Pi or a Pi to do first, or a zero again and the pattern will continue. So it has a same shape and structure as x cubed almost, but the evidence of 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. With all periodic functions, one of the questions you might want to ask yourself is, what is its period? How long does it take for it to repeat itself? Created inside the cosine is one lap around the inner circle of 2Pi. For tangent, the period is just Pi due to the quotient nature of tangent. So this is a nice graphs to know. It's good to have the general shape of this graph. It does come up enough, not as much of course. The sine and cosine, let's say those two are the most that are used the most often, but tangent is another good one to know. So now, let's do an example. 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 2_x. So I want you to understand how small changes to, this is the coefficient on x, what does it do to the graph of tangent? 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 x, y axis and I start plugging things in, I can ask myself, what is tangent of zero? You can check, once again, that you go through right to the origin at zero. You want to start thinking about, what is the relationship of sine of 2_x and cosine of 2_x? How does that change what this function was before? One thing you might want to do is ask yourself, 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 Pi over 4 or 3 Pi over 4. So you're shrinking these gaps, the period inside of tangent. So let's draw those here. So Pi over 4, 3 Pi over 4, and by symmetry, you can do it on the backend, negative Pi over 4 and negative 3 Pi over 4. Draw in your dashed lines to get your vertical asymptotes, and then draw the shape. Nothing else is really, there's no negative signs, we're not squaring anything. So the general shape is preserved, though what this does, this is the two in it, it changes where the vertical asymptotes lie. So then you draw your little disco functions in between, make sure you pass them right through the origin, and you draw a little more symmetric than 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 sine. If you really want it 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 to the calculator of what this means. It's like talking and 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, and we'll see some of the next video where I do this in Desmos, and you get to see how manipulations of the graphs occur. Real quick. Just the other graphs that I want you to just have a general idea of are going to be y equals cosecant x. We won't do any transformations of these, but we'll just get the general idea. Cosecant x is the reciprocal of 1 over sine. So of course, it's 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. It's height, of course, is 1 to negative 1. This is min and that's is max. Here's a good old sine of x. So what happens when you take a number and you flip it? For example, when I take zero at the origin, this function up at zero and flip it, I'm all of a sudden undefined. So everywhere sine of x is zero, cosecant x will be undefined. You 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 its max of 1, 1 over 1 is just 1. So wherever you have a peak or a valley, the cosecant function will match at this point. Then you have to realize just the relationship between numbers in general. As numbers get smaller, their reciprocal gets bigger. Think about that for a minute, or vice versa, as numbers get larger, the reciprocal gets smaller. So maybe that's the 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-seventh, I'm getting smaller. With that property, you can then fill in the general shape of this graph. As sine gets smaller, heads towards the origin, it's graph gets bigger. So as it goes down, the graph of the cosecant goes up, and then vice versa. Once you realize that pattern for reciprocals, you can then draw this shape. With any periodic function, you'd like to know its period. So since this is just the reciprocal of sine, its period as well is 2 Pi. Once you understand or see how cosecant behaves, then finding the graph of secant is very similar. This is 1 over cosine of x. So let's draw the parent function cosine of x first. Cosine of x is like horizontal shift of sine, it starts at 0, 1, it goes up, it goes down, maxes out a one because it is 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 where this function is no longer defined, cannot take the reciprocal of zero. Then where the function is one, it's going to match, at negative one, it's going to match. So now we start doing the same thing, as the numbers get smaller, their reciprocals get larger and vice versa, and off it goes up, down and up, and we can start to build here the shape of cosecant. If you can imagine in your mind now deleting the graph of cosine, deleting graph of sine from this picture, you get these Us that alternate with period 2 Pi once again. So I'll write that down. So period is 2 Pi. The reciprocals are 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 I haven't quite grasped sine and cosine just yet. So you really want to make sure you have a good handle of sine and cosine. That means knowing the graph, knowing some values, knowing how to find values in different quadrants, knowing the sine, 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 into 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.
