Hi, everyone. 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 hand you packets of papers, and identities, and it looks mildly terrifying. You get this kind of picture of the unit circle with all four quadrant labeled. There's serious stuff in here. There's pies, 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, and then I can derive the rest. I don't need to memorize this one. 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 this stuff, and you have a minute to fill in the boxes, or fill it all in. It was terrifying, it was stressful, it was no fun. Nowadays, we use a calculator. You can look stuff up online. You don't need to have things memorize. It's better to focus on the understanding, so that if there is a deviation from a question, or some 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, or 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 1. There's a couple of things in quadrant 1 that we got. Pi over four is half the square, pi over six, and pi over three are half the equilateral triangle. We understand the unit circle, everything's in quadrant 1. Then if they ever ask us about something in quadrant 2, 3, or 4, we just use the reference angle, and relate it back to something in quadrant 1. I don't have this memorized. I always solve it if I need to, and you can always look it up, if you need to, as well. Teaming 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 montage in the movie, where this stuff is flying around, and it's awful. Watch out for this. Ignore this. We're going to do in this lecture is, I'll show you the core identities, the basic identities that you need to know. Then if you ever need the rest, you can look it up, or you could even understand, and derive it for free. 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 exists, but we never use them for anything. So in the off chance that you need them, just go look it up. No problem. I just want you to understand them, know the functions are, and I don't want this to be you, some stock photo inside an oval pulling your hair out. So yeah, that shouldn't be you, but 'yey' for stock photos. Enough of that. One more time, the idea of this course that I want you to take away from this: don't feel like you have to memorize this stuff, don't call my flashcards. All the trig functions, all the periodic functions are related to each other. They're all defined in terms of coordinates on the unit circle. Keep that in mind. For this reason, any expression involving these periodic functions, can be written in many different forms, depending on what quadrant you're in, depending on how you want to write it with the 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. Their use is to simplify expressions, determined whether expressions are equivalent, or recall that identity, also, could be satisfied by all different numbers. We want to find solutions to these identities. We're going to focus on the basic ones. Friendly reminder, just build a strong foundation, as you can see here, and the rest will be the lesser we gain. 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. These are called our even and odd identities. Just going to list these because we have some already. Even and odd identities. Friendly reminder, what they are. Remember an odd function, let's 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. It's named from odd because three is of course odd. Friendly reminder, y equals x cubed, this is our disco function. Starts off low, goes high, it goes right through the origin. This thing has rotational symmetry, or origins symmetry. 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 negative of some number, you get back the negative of the output of a 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, friendly reminder, though hopefully, this is all coming back to you, is one that looks like 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 one, or negative one, I square it, I get the same output. This was one where we had f of negative x is equal to the same output as f of x. So the minus sign goes away. 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 you will get for free. Sine, sine starts at the origin, it goes up and down with a period of two Pi. Its maximum, of course, is one, and it's minimum, of course, is negative one. It does it both to the right and to the left of the y-axis. Maybe we'll go one full period to the left. This is not the greatest drawing, 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 fold this over on itself, sorry, 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 origins symmetry. It is an odd function. 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 you simplify the algebra. If you plug in a negative, then you get positive here. For the other one, f of x is, of course, cosine of x. Now, think of the graph of cosine, let the symmetry be your guide. Cosine we'll start at the point 0,1, we'll go down and then up, and we'll do that forward and backwards. We'll do one period to the left, to the right of the y-axis. 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 can 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. These are called our 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'm going to tell you a little story here. When I was debating, I'm 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 like one or two years from retirement. He's a very soft-spoken, short guy, and super nice and just liked to tell stories. One time I asked him for math help and he said, "Have you tried staring at the Moon?" That was this his legitimate thing. I was like, "No, I have not tried that". He said, "We'll 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, like just another story from this guy. I used to do work too fast, do a lot of stuff by 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 was these little things that just make you stop and go,"Oh,okay". Again, I remember these stories. One thing that he always said, maybe it's a little outdated or whatever, but he's like, "cosine is strong". The guy was like cosine is strong. Cosine, it can handle, it can absorb the negative sine, and I always remember this, "Cosine is strong, sine is gentle". He'd even say cosine is the father and sine is the mother. Every math problem is your baby. He's like cosine is the father and he'd say, "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 the standards of whatever, but like it's memorable, I don't know. Take it for what it's worth. To this day, when I look at these things, I say cosine strong, sine weak, gentle, I bring it over. I gently carry that minus sine and make the motion with my hands as if I'm been bringing out a small child holding it in my arms. One thing, however you remember things for whatever reason, many, many years later that has stuck 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 that even or odd? So even or odd. Now the graph of cosecant like yes, 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 and cosine, your first approach, hey, 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 if 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. So the question is, what's going to happen to this negative, does it go away or does it carry out front? This becomes one over sine of negative x, and then we gently, like a mother carrying a baby, move it out. There you go. Don't wake up the baby. Negative sine of x and this becomes negative. We could bring that negative out front. Sine of x. Hey, wait a minute, 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. Cosecant, in fact turns out to be odd. This is where like I don't know about cosecant, but I can definitely like plug these in and just keep bringing it back to sine and cosine. Once you know everything about sine and cosine and 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 and 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 carried out front, negative sine of x. Cotangent, this is one over tangent. This is also sine over cosine is reciprocals. This would be like cosine of negative x over sine of negative x. That's what cotangent is and I have cosine, everything's back in terms of sines. This is always a good strategy to use because then you can know your foundation and the rest follows accordingly. Cosine is strong, negative goes away. Cosine, sine gently move the negative out. Don't like the baby. Minus sine of x, the two negative signs go away, and in fact, the sine functions cancel. This simplifies quite nicely to cosine of x. Another set of identities are called the Pythagorean identities and if you know the Pythagorean formula, a square plus b squared to c squared. I promise you will know this. We call this the fundamental identity because it is the foundational one you need to know. Friendly reminder, let's draw the unit circle on the xy plane. Let's draw some angle Theta off of the x axis where we 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. 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. There's always a triangle associated to any angle off the x axis and let me just, now I'm going to redraw the picture just a little larger so we can talk about it. I'm going to redraw that right triangle. Remember the unit circle guarantees hypotenuse is one, your x is how far over you are, and your Y's how far up you are. We call x's the cosine due to SOHCAHTOA. 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 content. Look at the name of slide by Pythagorean formula. The Pythagorean formula says that x squared plus y squared is equal to one. One squared is one, I guess we'll just call one. You get the x-squared plus y-squared is one, that side squared plus this side squared is hypotenuse squared. In the land of trig and the land of periodic functions, x is cosine. This becomes cosine squared of Theta plus sine squared of Theta is one like 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. Again, 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 like you got to commit to memory at some point. It is like the core of the foundational thing. If you're keeping a little formula sheet thing at home, again, don't download one off the Internet so only 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. Let's, how did we get those? One thing we can do to an equations we can manipulate it using algebra. Let's divide everything by, let's do cosine squared of Theta. What happens when I do that? The cosine squared Theta plus sine squared Theta equals one, 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 and choose whatever you do to one side, you must do to the other side. Here we go. Cosine squared divided by a cosine squared, of course is one, they cancel. We get one. Sine over cosine, that's tangent of Theta and everything squared. We'll just do a tangent squared and then one over cosine squared well, this is the reciprocal of cosine. This is of course, secant and we have our second, is 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 1, 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, we start with identity again, cosine squared plus sine squared Theta is 1. There's 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. 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. The numerator is equal to the denominator. Then 1 over sine squared, that's the reciprocal of sine, whole thing is squared. That's good old cosecant squared. This one does not come off as much, but it's perfectly fine, again, because I know the process, I can always just go off on the side for two minutes or less maybe, 30 seconds, 10 seconds, and go find these things. We get three identities. It's cosine squared plus sine squared is 1 is your foundation. 1 plus tangent squared Theta plus equals secant squared, Theta is your second variation where the first variation of it and then cotangent squared Theta plus 1 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 where the second too 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. We'll put equals sine, I like, we'll keep, cotangent, I don't like. Cotangent is the reciprocal tangent. This is cosine of x over sine of x. Then we have a cosine of x floating around over here. Since we're dealing with fractions, like maybe we write as cosine over 1. 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, I put the cosines together. I have cosine squared of x over sine of x. Wonderful. Zoom out for a minute. See there from space. I have fraction plus a fraction. How do we add fraction plus a fraction? It will put a one here to stress that well, we went to elementary school. Got to use common denominator. Let's do sine times the top and times the bottom for both. 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'll write this expression over a common denominator, and I will then add the numerators to get sine square root of x plus cosine squared of x. Well, do you see what I see, sine squared plus cosine squared. Hey, that's the one thing that we got a no, that is our basic identity. That becomes 1 and I get 1 over sine of x is the reciprocal of sine, better known as cosecant, csc of x. This gross expression with three functions can be cleaned up quite nicely to just one function. How nice is that? I will see another example along the same flavor. Let's practice our fraction work. We propose use review with that. Let's simplify the expression 1 over sine squared minus 1 over tangent squared of x. I have fractions, I have tangents. We don't like tangent, I rather everything is sine and cosine. Remember your basic strategy, if you want to go here is to simplify. Pause the video, try to simplify this as you go and let's play around this for a minute. Following, if you're ready, did you pause the video? Pause the video, try it on your own. I'll wait. Let's go. Tangent. I don't know. I want to convert that. I don't like that. We have 1 over tangent, 1 over sine squared. Let's write this as, we'll keep it in sine squared. No problem. Tangent though not in sine and cosine. Let's do that friendly reminder. Tangent, tangent of x is sine over cosine. We remember that from SOHCAHTOA. We have 1 over and now it's in the denominator and everything's flips so going to do is a one-shot. I want to take the reciprocal. So that turns it into cosine x over sine of x. Because it's 1 over so I do like the reciprocal here. You got to flip it and then everything is squared. Just don't want to forget that. Don't want to forget that. What's nice about that when you clean it up is that you have 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, you had to have a common denominator so you can just subtract the numerators, 1 minus cosine squared over sine squared of x. Now 1 minus cosine squared. 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. One minus cosine squared x becomes sine squared of x. You can always set equation. You can move things around 1 minus cosine squared, that becomes sine squared of x. There is our fundamental identity in action once again. I get sine squared over sine squared of x. Hey, wait a minute, numerator equal to denominator, that of course, is equal to one. Who saw that coming? If you've got a one there, great job, wonderful job. Let's do another one. Secant Theta minus 1, secant Theta plus 1. This is a little different. These are not fractions, but I have something times something so you want to multiply this out. For the reminder, when you multiply things out like this, you have to foil. We do first outside, inside last. Here we go. First, the secant Theta times secant Theta, that of course is secant squared Theta multiply the two together. Outside says 1 times secant, that is secant Theta. Inside it says negative secant Theta, oh, those will cancel on the second. Then foil first outside, inside, the last gives you, of course negative one. The plus secant Theta, the minus secant Theta, we say goodbye to those and we get secant squared Theta minus one. This is already a little nicer. It's now 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. If I move things around, secant squared Theta minus 1 is tangent squared Theta. We are going to use that secant squared Theta minus 1, this just becomes tangent squared. It's 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. 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. Let's first off. Usually, when you have some of it ugly, let's clean it up and make it all pretty. We can turn this tangent into sine and cosine. That might not be a bad way to go, so let's follow our strategy. This becomes cosine of sine Theta over tangent Theta. Whoops, I messed up. Try that again. What is tangent Theta's? 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. 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. Now I have sine Theta plus 1 times sine Theta minus 1 all over cosine squared of Theta. Beautiful. Numerator looks familiar, we just had a problem like that. We're going to foil this guy out again. Here we go. First sine times sine is sine squared. Outside, negative sign. Inside, positive sign. First outside, inside and lasts puts a minus 1 all over my denominator of cosine squared. The negative sine and the positive sines they cancel, 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. What can I write? How can I rearrange this to look like sine squared minus? Well, I can move the one over and I can move negative cosine squared to the other side as well. Don't be afraid to manipulate your identities. When you do that, you're 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 one as your final answer. This is kind of really scary gross expression for any value of Theta will always equal negative one. 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 one for every single value. If you've got that one, that's a fantastic job. Try some more of these on your own. Keep the formulas handy, and I will see you next time.
