Hi, everyone, and welcome to our lecture on some basic identities using all the periodic functions that we've seen before. Now, this section always gives people a little bit chills and the stuff that nightmares are made of. Because if you do a quick Google search or if you've ever seen this stuff before, they usually hunch packets of papers, and identities, and it looks miley terrifying, like you get this picture of the unit circle with all four quadrant labeled. There's scary stuff in here, there's Pis, and there's weird angles, and everything, but if you think about the approach that we've been taking, I want you to basically ignore this. I want you to just focus on, what are the core things I need to know? Then I can derive the rest. I don't need to memorize this. I never like the way that I was taught this, I didn't like it when I was taught this. We used to have speed quizzes where they would just make you memorize these stuff and you have a minute to fill in the boxes or fill it all in, and it was terrifying. It was stressful, it was no fun. Nowadays, like we use a calculator, you can look stuff up online. You don't need to have these memorized. It's better to focus on the understanding so that if there is a deviation from the question, verses slight variation of a graph, you understand where the pieces are going. So please take that with you that I'm never going to ask you to memorize stuff for speed stuff. You are allowed to look stuff up all the time. It's more about the understanding. For the unit circle here alone, this picture is terrifying, ignore it. Just focus on quadrant one, there's a couple of things in quadrant one that we got. Pi over 4 is half the square, Pi over 6 and Pi over 3 are half the equilateral triangle. We understand the unit circle, everything's in quadrant one. Then if they ever ask us about something in quadrant two, three or four, we just use the reference angle and relate it back to something in quadrant one. I don't have this memorized, I will solve it if I need to. You can always look it up if you need to as well. Seaming is going to happen for identities. If you lookup identities, this is the first Google search I got. There are pages and pages of terrifying things. You can imagine the mountain tops in the movie where there's stuff's flying around and it's awful. So watch out for this, ignore this. What we're going to do in this lecture is show you the core identities, the basic identities that you need to know and then if you ever need the rest, you can look it up or you could even understand and derive it for free. So some of these we'll go over and some of these we'll never go over because honestly, they don't come up. Yes, they exist, but we never use them for anything. So in the off chance that you need them, just go look it up, no problem. We just want to understand them, know the functions are. I don't want this to be you, some stock photo inside an oval pulling your hair out. So that shouldn't be you, but here is for stock photos. Enough for that. This is one more time the idea of this course that I want you to take away from this. Don't feel like get the memorizing stuff, don't go and make flashcards. All the trig functions, all the periodic functions are related to each other. They're all defined in terms of coordinates on this circle. Keep that in mind. For this reason, any expression involving these periodic functions can be written in many different forms depending on quadrant you're in, depending on how you want to write it with reciprocals or not. Again, bring things back to sine and cosine. Identities, these are going to be equations, these are things where we equate one thing to another. They're used to simplify expressions, to determine whether expressions are equivalent, or recall that identities also could be satisfied by all different numbers. So we want to find solutions to these identities. So we're going to focus on the basic ones. Find the reminder, just build a strong foundation as you can see here, and the rest will be okay. So I will call out what I want you to know, and we're going to start with some of the identities that we already had before, and these are called our even and odd identities. Just going to list these because we have some already. So even and odd identities, find the reminder what they are. Just remember an odd function, we'll start off with an odd function, an odd function is one that looks like y equals x cubed or behaves like y equals x cubed. Is named for odd, because 3 is of course odd. For the reminder, x cubed is our disco-function, starts off low, goes high, goes right through the origin. This thing has rotational symmetry or origin symmetry. You imagine grabbing the handles on the top left and bottom right, bottom left but top-right, and it's giving you the spin, and it will land right on itself. Algebraically, this means that if you plug in the negatives of some number, you get back the negative of the output of the positive number, f of negative x equals negative f of x. This is the algebraic way to show that something is in fact, an odd function. An even function, for the reminder, hopefully this is all coming back to you, is one that has the same symmetry as y equals x squared. The parabola, this is a function that has y-axis symmetry. This is where if I plug in 1 or negative one and I square it, I get the same output. So this was one where we had xf of negative x is equal to the same output as f of x. So the minus sign goes away. So let's look at our odd function first, our odd functions is going to be the graph f of x equals sine of x. Always study sine and cosine, the rest we will get for free. Sine starts at the origin, it goes up and down with a period of 2 Pi. Its maximum of course is one and it's minimum, of course, is negative one, and it does it both to the right and to the left of the y-axis. So maybe we'll the wonderful period to the left. This is not the greatest strongly of course, but it's supposed to be beautifully symmetric about the y-axis. You can imagine if I had drawn this perhaps a little better. If you rotate this thing about the origin, so grab the handles, give it a spin, 180 spin, it'll land right on itself and has origin symmetry, so it is an odd function. So sine of x is an odd function. If I take sine of negative x, this becomes negative sine of x. Recognizing what the graph is will help simplify the algebra. You plug in a negative, then you get positive here. For the other one, f of x is of course cosine of x, and I think of the graph of cosine, let the symmetry be your guide. Cosine will start at the 0,1, will go down and then up, and we'll do that forward and backwards. We'll do one period to the right of the y-axis, and in that case here you can imagine folding this thing in half over the y-axis. This graph exhibits y-axis symmetry, and in that case, it is an even function. Just like the probability you could fold it over, cosine graph you could fold it over as well. The identity that we get out of that is that if cosine of a negative is input, we will get back the same as cosine of x. So our two, these are called even and odd identities. The two things to realize is that cosine of a negative it doesn't matter and that sine of a negative, the negative pulls up front. Once you know these two things, the rest you get for free. I'll tell you a little story here, and I was debating on telling you this or not because who knows, but it wasn't me who said it. When I took this trig and periodic functions and all these thing for the first time, I had an older Japanese teacher from Japan and he was maybe one or two years from retirement. So very soft-spoken, short guy, and super nice, and like to tell stories. One time I asked him for Math help and he said, "Have you tried staring at the moon?" This is legitimate thing. I said, "No, I have not tried that." He said, "Well, try that and then come back." I don't know how helpful it was, but it was especially memorable for sure. One of the things that he would tell and tell stories about, just another story from the sky. I used to do work too fast, to do a lot of stuff far head, and he'd be like, "Sometimes it's bad to drive a Ferrari, when you go too fast you miss the flowers on the side of the road." Anyway, it's these little things that just make you stop and go. Again, I remember these stories. One thing that he always said, and maybe it's a little outdated or whatever, but he was like, "Cosine is strong." He's this very the guy who got cosine is strong, it can absorb the negative sign. I always remember this, "Cosine is strong, sine is gentle." He did even say, "Cosine is the father and sine is the mother." Like every math problem is your baby. He said, "Cosine strong, cosine kill the negative. Sine, like mother, gently bring negative up front." I know it's a weird story and perhaps not up to today's standards of whatever, but it's memorable. So take it for what it's worth. Till this day when I look at these things, I say, "Cosine is strong. Sine, weak, gentle." I might bring it over and I gently carry that minus sign and make the motion with my hands as bringing out like a small child holding it in my arms. However you remember things for whatever reason, many, many years later, that has stuck and as we go through it. Keep those in mind and let's see how we're going to use these things. Let's take cosecant, remember this is the reciprocal of sine. Is it even or odd? Now the graph of cosecant, maybe you'll remember it, but it doesn't come up that much. Let's use the algebra and then you can go check it with the graph. You get a question about a function that's not sine or cosine and your first approach, let's put this back remember it's the middle letter we use, let's put this back in terms of sine. I'm really thinking of this as the reciprocal of sine, and is it even or odd. Let's see CSC, let's check. Let's throw a negative n. The question is, what's going to happen to this negative, does it go away or does it carry out front? So this becomes 1 over sine of negative x, and then we gently, like a mother carrying a baby, move it out. There you go, now we got the baby, Negative sine of x and this becomes negative, we could bring that negative out front, sin of x, this is negative cosecant of x. The negative has in fact come out front. It's come out front, and when that happens, just like x cubed, you have an odd function. So cosecant in fact turns out to be odd. So this is where like, I don't know about cosecant, but I can definitely plug these in. Just keep bringing it back to sine and cosine, once you know everything about sine cosine, you just bring it back and you have a strategy every time. I'll give you an expression and we're going to simplify it. How do we do sine of negative x and then cotangent of negative x. Cotangent, not sine and cosine, sine, it is sine, so we're good. Again, what's the strategy for this thing? Well, let's turn right away. Well, I guess we could do it in one step. Sine of negative, remember the negative we gently carry it up front. Negative sine of x. Cotangent, this is one over tangent. This is also sine over cosine's reciprocal. This would be cosine of negative x over sine of negative x. So that's what cotangent is. I have cosine, so everything's back in terms of sine. This is always a good strategy to use because then you can know your foundation and the rest follows accordingly. Cosine, cosine is strong, negative goes away. Cosine. Sine, gently move the negative out, don't wake the baby, minus sine of x. The two negative signs go away, and in fact, the sine functions cancel. This simplifies quite nicely two cosine of x. Another set of identities are called the Pythagorean identities. If you know the Pythagorean formula, A square plus B squared is C squared, I promise you will know this. We call this the fundamental identity because it is the foundational one you need to know. A friendly reminder, let's draw the unit circle on the x, y plane. Let's draw some angle Theta off of the x-axis wherever you want it to be, and let's drop a line down so we form a right triangle, and we'll have our value x, y right on the unit circle, right where it touches. So x is how far you're over, and y is how far you're up, and on the unit circle, you have a nice radius of one. So there's always a triangle associated to any angle of the x-axis. Now I'm going to redraw the picture just a little larger so we can talk about it. So I'm going to redraw that right triangle. Remember the unit circle guarantees the hypotenuse is one, your x is how far over you are, and your y is how far up you are. We call x is the cosine due to SOHCAHTOA. So cosine of your angle Theta and y is sine of your angle Theta. This is at the end of the day, a right triangle. If you're going to know one formula about right triangles, what should it be? Hint, look at the name of the slide, Pythagorean formula. The Pythagorean formula says that x squared plus y squared is equal to one, one squared is one, I guess, but just call one. So you get the x squared plus y squared is one. That side squared plus this side squared is the hypotenuse squared. In the land of trig, in the land of periodic functions, x is cosine, so this becomes cosine squared of Theta plus sine squared of Theta is one. This is something that you absolutely have to know. We've encountered this previously. This is the basic Pythagorean identity. It doesn't matter what Theta is. I'm using Theta here because I'm calling the side x. If you talk about functions, these are dummy variables, Theta or x are perfectly fine. This is one that you got to commit to memory at some point, it is the core or the foundational thing. If you're keeping a little formula sheet thing at home, again, don't download one off the Internet, some of that, they're terrifying, just keep a little handy dandy thing in the back of your notebook somewhere, this is one that's got to be there. Now, I claim if you know this one, you get two more for free. How did we get those? So one thing we can do to an equation is, we can manipulate it using algebra. So let's divide everything by, let's do cosine squared of Theta. So what happens when I do that? Your cosine squared Theta plus sine squared Theta equals one. So let's start with our identity. Here we go, we're going to divide everything by cosine squared theta, why? Because I can, and I want to. As long as I don't break any rules of algebra, it's perfectly legal, and the key is just do it to every single piece. You can't, just like pick or choose, whatever you do to one side, you must do to the other side. Here we go. Cosine squared divided by cosine squared, of course is one, they cancel, so we get one. Sine over cosine, that's tangent of Theta, everything squared, so we'll just do tangent squared. Then one over cosine squared, well, this is the reciprocal of cosine. This is of course secant, and we have our second Pythagorean identity. This is a variation of the first, and this is one that comes up enough. Do I have it memorized? Well, only from doing it 1,000 times, but honestly, once in a while I'll still check. The thing that I want you to understand is how did you get here? If you have your cosine squared plus sine squared is one, you should know you can get two others, and that you do a simple division, just divide everything by cosine squared. We're going to keep this approach to get our next one and our last one. We can also divide everything by sine squared. When we do that, I start with Identity again. Cosine squared plus sine squared theta is 1. These are our foundational basic identity. Divide everything by sine squared. Why? Because I can, and because I want to, and because it's perfectly legal, and not breaking any rules. So here we go, cosine over sine, that should look familiar. What is that? That's good old cotangent. So we'll do cotangent squared. Sine over sine, of course they cancel to get one, numerator is equal to the denominator. Then one over sine squared, that's the reciprocal of sine, whole thing squared. So that's equal to cosecant squared. This one does not come up as much, but it is perfectly fine again, because I know the process. I can always just go off on the side for two minutes, unless maybe 30 seconds, 10 seconds, and I go find these things. So we get three identities and it's cosine squared plus sine squared is one is your foundation. One plus tangent squared theta plus equals secant-squared, theta is your. Second variation or your first variation of it and then cotangent squared theta plus one equals cosecant squared, there's your last one. These three tend to get lumped together. These are called the Pythagorean identities. Know the first one and then know here the second two come from. Let's do an example. I'd like to simplify sine of x plus cotangent x cosine of x. Scary expression, three functions, no [inaudible]. Now, cotangent. Remember the strategy. I know everything about sine and cosine. I'm not going to memorize the entire stuff. I want things in terms of sine and cosine. No problem. So we'll put equals sine, I like, I'll keep. Cotangent, I don't like. So cotangent is the reciprocal tangent. This is cosine of x over sine of x, and then we have a cosine of x floating around over here. Since we're dealing with fractions, maybe we write as cosine over one, it doesn't matter. But the point I want to look at here is that I have when I combine the fractions on the right or I put the cosines together, I have cosine squared of x over sine of x. Wonderful. Zoom out for a minute, see the earth from space and I have fraction plus a fraction. How do we add fraction plus a fraction? I will put a one here to stress that. Well, we went to elementary school, we got to use common denominator. So let's do sine times the top and times the bottom for both. So we have sine squared of x over sine of x, common denominator. Cosine squared of x over sine of x. I have my beautiful common denominators. So I will write this expression over a common denominator and I will then add the numerators to get sine squared of x plus cosine squared of x. Well, do you see? What I see, sine squared plus cosine squared, that's the one thing that we got to know, that is our basic identity. That becomes one and one get one over sine of x is the reciprocal of sine, better known as cosecant, CSC of x. So this gross expression with three functions can be cleaned up quite nicely to just one function. How nice is that? Let's do another example along the same flavor. Let's practice our fraction work. Because we use review of that. So let's simplify the expression, one over sine squared minus one over tangent squared of x. So I have fractions, I have tangents. We don't like tangent, I rather everything in sine and cosine. So remember your basic strategy. If you want, the goal here is to simplify. Pause the video, try to simplify this as you go and let's play around this for a minute. So following if you're ready, please pause the video, try it on your own. I'll wait. Let's go. Tangent, I want to convert that. I don't like that. So we have one over tangent, one over sine squared. Let's write this as we'll keep it with the sine squared, no problem. Tangent though not in sine and cosine. So let's do that further reminder. Tangent of x is sine over cosine. So we remember that from SOHCAHTOA. So we have 1 over, now it's in the denominator and everything is flip. So I'm just going to do this in one shot. I'm going to take the reciprocal. So that turns it into cosine x over sine of x. Because it's one over, so I'm going to do like the reciprocal here. You got to flip it and then everything is squared. So I just don't want to forget that. What's nice about that when you clean it up is that you're in fraction minus fractions, so big picture fraction minus fraction, they already have a common denominator. How nice is that? In the first one, they don't have a common denominator. But when you clean it up and write it in terms of sine and cosine, this is always a very good strategy to have. They have a common denominator, so you can just subtract the numerators. 1 minus Cosine squared over Sine squared of x, and now 1 minus Cosine squared. So remember, you should have this thing handy, keep it easy to reference. We have Sine squared plus Cosine squared of x is 1. Do a little rearranging, 1 minus Cosine squared x becomes Sine squared of x. So you can always set equation, you can move things around, 1 minus Cosine squared, that becomes Sine squared of x. So there is our fundamental identity in action once again, and I get Sine squared over Sine squared of x. Hey, wait a minute, numerator equal to denominator, that of course is equal to 1. Who saw that coming? If you've got 1 there great job, wonderful job. Let's do another one. Secant Theta minus 1, secant Theta plus 1. Now this a little different. These are not fractions, but I have something times something, so you want to multiply this out. Further reminder, when you multiply things out like this, you have two foil. So what you do, first outside, inside last. Here we go. First, the secant Theta times secant Theta, that of course is secant squared Theta, multiply two together. Outside, says 1 times secant, that is secant Theta. Inside, it says negative secant Theta. Oh, those will cancel on a second, and then foil first outside, inside last gives you, of course, negative 1. The plus secant Theta and the minus secant Theta, we say goodbye to those, and we get secant squared Theta minus 1. This is already a little nicer, it's not two things multiplied together. In the original expression, we had four terms, now I only have two. But of course, where am I going with this? Secant squared Theta minus 1, that looks like something we can do. Remember the Pythagorean identity, where is it? It's back here somewhere. Secant squared, if I move things around, secant squared Theta minus 1 is tangent squared Theta. So we are going to use that, secant squared Theta minus 1, this just becomes tangent squared, and this is nicer to write it in terms of a single function, why not use your identity, keep them handy. Remember, you're allowed to move things around if you need to as you go. Now, let's do one more example. Let's make it a little harder, if we could do this one we could do any of them. Here we go. Let's try, Cosine of Theta times tangent of Theta plus 1, Sine Theta minus 1 over Cosine squared Theta. Let's put it all together. Here it is, folks. We've got things multiplied together and fractions. This is exciting. Are you ready? I'm excited. Usually, when you have something ugly, let's clean it up and make it all pretty. We could turn this tangent into Sine and Cosine, that might not be a bad way to go. So let's follow our strategy. So this becomes Cosine of Sine Theta over tangent Theta, I messed up. Try that again. What is tangent Theta? So I was thinking about tangent and I wrote tangent. Sine over Cosine, there we go. We're all good. Put that in the first one. Then we have Sine of Theta minus 1, and I don't want to lose my denominator, so I'll rewrite that as Cosine squared Theta. Why do I write tangent as Sine over Cosine? Look at that Cosines cancel. So now I have Sine Theta plus 1 times Sine Theta minus 1, all over Cosine squared of Theta. Beautiful. Numerator looks familiar. We just set a problem like that, we're going to foil this guy out again. So here we go. First, Sine times Sine is Sine squared. Outside, negative sign, inside positive sign. So first, outside inside and last, puts a minus 1 all over my denominator of Cosine squared. The negative Sine and the positive signs they cancel, see you later, and I have Sine squared minus 1. Now, hopefully you're seeing the pattern on these things. Think about your fundamental identity, we have Sine squared Theta plus Cosine squared Theta equals 1. How can I rearrange this to look like Sine squared minus 1? Well, I can move the 1 over and I can move negative Cosine squared to the other side as well. So don't be afraid to manipulate your identities. When you do that, your Sine squared minus 1 becomes negative Cosine squared of Theta, all over Cosine squared of Theta. Cosines Theta of course, cancel, and you're left with a very beautiful negative 1 as your final answer. So this is really scary gross expression, for any value of Theta will always equal negative 1. That's amazing. Try that if you want, plug some values in the calculator, radians or degrees, it does not matter. As long as you use your parentheses carefully, you will get minus 1 for every single value. If you've got that one, that's a fantastic job. Try some more of these on your own, keep your formulas handy, and I will see you next time.
