Hi everyone. Welcome to our lecture on sine and cosines, more about sine and cosine. We're still going to treat them now as functions. It's a very similar thing, but we're going to leave the unit circle is the idea. So our goal for this section will be to leave the unit circle. So what does that actually mean? So before, everything we have is on the xy-plane, we draw the unit circle. It's not my best circle I've ever drawn, but we'll go with it. So of course, it has radius one. The idea is that if we draw a triangle with the vertex of the origin, and whose hypotenuse is the radian, it forms an x and a y as its adjacent and opposite sides. We create a little angle Theta, and we use the fact that y, the sine of Theta was y and cosine of Theta was the x-value of the point, where this triangle intersected the unit circle, and that's all great. But what's going to happen now as we move on to do bigger and better things is, we're going to look at triangles where the hypotenuse is not one. So you can imagine if you're trying to measure some buildings or do some architecture, you're doing whatever modeling you want to do. Well, your triangle may not be of size one, and so you'll start to see word problems where, you draw a triangle and the given information is just larger. So for example, the 3-4-5 triangle. So what do we do in this case? Well, in this case now, if you have some particular point and you're looking at it in terms of its x and its y. So at the vertex of this triangle, so x is 3 and y is 4. What we can do to preserve ratios is I want to look at the new hypotenuse. What is the new hypotenuse? We call this r, corresponding to the radius of some circle. Now, if you notice in this case, the triangle I drew, the 3-4-5, the radius is five. So if I were to embed this triangle at the origin in Quadrant 1 at around a circle. So you can imagine some circle there that touches the triangle at its point 3,4. Well, it's not the unit circle, the radius is clearly 5. So you can find this radius, of course, in general, just using the Pythagorean theorem, you see this a lot. If I give you the point x,y. Well, this is just good old x squared plus y squared, all under the square root. So r equals the square root of x squared plus y squared, and in that general case, you may not get 1. So may not equal 1, and that is okay. Not every triangle lives inside of the unit circle, that's perfectly fine. In that case though, that's not a problem. We go back to good old sohcahtoa. If we write out sohcahtoa, I'll just do it for sine and cosine. So, S-O-H, sine is opposite over hypotenuse and cosine is adjacent over hypotenuse. How does this change our formula? So sine of Theta, which is normally opposite over hypotenuse, becomes opposite over hypotenuse. Well, in our triangle here, that's not the unit circle. Since our hypotenuse is not one anymore, I can't just call it y. I have to call it the y-value over the hypotenuse, which is r. R again coming from this imaginary circle that we're embedding the triangle into, and cosine is exactly the same. So cosine of Theta will become our adjacent over hypotenuse, and that becomes x over r, Okay? So these are the most general formulas for sine and cosine. So we're generalizing our functions here, and you will get the same answer. That's not going to change, as long as you're consistent. But just realize the nice way of looking at the triangle and grabbing the x and y-value that we had before in previous videos, sine of Theta is y, cosine of Theta, this is only for the unit circle. If you leave the unit circle, you need to put divided by its hypotenuse, which is no longer one, and so now this is for any point. The beauty of course as we saw before, these are all similar triangles, and so the values are preserved. If you want to torture yourself, go google trig identities or identities involving sine and cosine. There are pages and pages, and pages of them. If you want to torture yourself, sure, go ahead and do that. I say, just know one of them, just know one thing. So I keep comparing math to cooking. Yes, there's a billion recipes out there, but you always have your one thing that you go to, the one thing that you do well. If we start talking about sine and cosine, and you're visualizing a beautiful right triangle, and you can call. Let's just do the general case. We have some hypotenuse r, I have my y-value and my x-value for my adjacent opposite sides. Again, you're imagining it in some sort of unit circle. In some sort of circle, maybe it's unit, maybe it's not, whatever you want. In that case we have that x, of course, x is the coordinate where the circle and the triangle intersect. So x is r cosine Theta, just rearrange some things around and y is r sine of Theta. See it this way as well. It's good to get used to doing it both ways. Again, if you want a picture on the unit circle, just replace r with one. That's not a problem. Then there's one identity that you have to notice. You all know I don't want to have to learn more identities. The good news is this is nothing new. You are dealing with a right triangle. I don't know when they start teaching it. It's pretty early, like middle school, elementary school. It's the one thing you have to know from a right triangle, what's the relationship between the sides. It's a pretty famous theorem. It rhymes with smashmogorean theorem. It's the Pythagorean theorem. Yay, of course. So the sum of the sides squares is hypotenuse squared. So x squared plus y squared is equal to r squared in our case. This is the Pythagorean theorem. I would like to trust the elementary education system that people have seen this before. I hope this is not something new like, whoa, I've never seen that before." So x squared plus y squared sometimes you see like a, b, and c, whatever the variables are, they're all dummy variables, who cares? But the side squared, sum them up is equal to the hypotenuse squared. In our case, we've been working with right triangles, so I always want to replace that with sine and cosine. So let's see what this turns into. Replace x with r cosine Theta, and you get r squared cosine Theta. So replace x with r cosine Theta and you have r cosine Theta squared plus replace y with r sine Theta, r sine Theta squared is equal to r squared. This is a little algebra here. So we're going to bring the two into both r and the cosine, hits both pieces and you get r squared cosine squared Theta plus r squared sine squared Theta. Same thing, bring the two in on the r and the sine that's all equal to r squared. Then divide every single piece by r squared. You can certainly do that. Whatever you do to one side, you must do the other. So we're treating things consistently and equally and you get cosine squared Theta plus sine squared Theta is equal to one. Notice this Identity if I had done with the unit circle, I would get the same thing. Cosine squared plus sine squared Theta is one. This is the fundamental, I'm going to call it the fundamental Pythagorean identity. This is the one identity that you need to know. If you know this, and again, we've been working with triangles, you should see where it's coming from. If you understand this, that's cosine squared plus sine squared is one, then you will be able to derive many, many other identities from it and you won't have to memorize anything. One quick little sidebar, come over here on the side with me for a second. Some warning about notation. When we square a trig function, a periodic function. So for example, you can write cosine Theta all in parentheses squared, but they don't do that for whatever reason. They write it as cosine squared of Theta. They put the two on the cosine because they didn't want to confuse anybody. Do you mean like am I squaring the Theta first and then taking cosine or am I taking cosine of Theta and squaring. So when you see me right the square on the Theta, just realize that it is like take cosine first and then square. Often when you don't have much going on, if it's just Theta, you don't write the parentheses. So this becomes cosine squared of Theta. The parentheses are implied. It's important to realize that this is not multiplication, this is a function, so we're taking cosine of Theta and then once we get that result, we are squaring the number. It's the same for any other periodic function. So sine squared of Theta is the same as parentheses sine Theta quantity squared. Let's do an example. Put all our good stuff to work here. So let's find cosine of Theta, if sine Theta is three-fifths and Theta is in quadrant two. So what does that mean? Again, when you get all this information, I always think it's best to draw a picture. Try to visualize, if you're a visual learner, see what's going on here. So Theta is sum Theta in quadrant two, we go counter clockwise when we label our quadrants. So Theta is somewhere back here. So we have some ray that ends up in quadrant 2 and we try to measure some angle Theta with its initial side, the positive x-axis and they tell me that sine of Theta is three-fifths. So right away when I draw the picture, I start to get the right triangle that appears. They don't tell me what circle I'm on. So why don't we, we can, well, I guess they kind of do. They tell me the sine of Theta is three over five. So remember, SOHCAHTOA. I'll write it all out. Sine is opposite over hypotenuse. So our hypotenuse, draw a right angle over here. This are opposite side. There's good old little triangle here is going to be three. Our hypotenuse is five, and I want the missing base of the sides. So maybe I'll come over and draw a new larger triangle so I don't try to squeeze it into my small picture here. So we have some other angle Theta, the thing that reference angle that we want. We have a right triangle. They're telling me that sine Theta is three-fifths. I do not know what Theta is. Do not know what Theta is, I just know it's ratio of the size of the triangle is three-fifths, the opposite side is three, and the hypotenuse is five. Given two sides of a right triangle, you can always solve for the other side. So you do five squared, you call it the x if you want. 5 squared is x squared plus 3 squared. You may recognize these numbers already, it is the 3-4-5 triangle. Let's go through the algebra. So you get 25 is x squared plus 9. Subtract 9 to both sides, x squared is 16, and of course x is 4. The algebraists might be yelling at me when I take a square root, you're supposed to plus or minus. This is the thing you're supposed to do. But since this is the measure of a triangle, I'm going to take the positive side. So x equals 4. So I have a 3-4-5 triangle. That triple comes up often enough for triangles here. Now, the key here is that I'm not only going to circle, the radius is 5, so we know x is 4. They're asking for cosine of Theta. Cosine of Theta, remember SOHCAHTOA. Cosine of Theta would be adjacent over hypotenuse. The length of the adjacent side is what I just found. That's 4. The length of the adjacent side is 4 and the hypotenuse is 5. So cosine of Theta is four-fifths, but you got to remember I'm using the reference angle. I'm using the helper angle to find this piece. I just want to check something here. The cosine of the value, remember, cosine of Theta is like your x value. I'm in Quadrant 2. Whatever this x value is over here, I am negative. I am very negative. The first piece of information, sine of Theta is three-fifths, will give you what the ratio is just because you can use Pythagorean theorem to find the missing third side. The second piece of information will tell you if it's plus or minus. Quadrant 2, the x value is negative. This thing is the final answer here, the cosine value that I want, because I'm in Quadrant 2, negative four-fifths. Final answer there. We start to treat the sine or cosine as a function on the unit circle, a pattern starts to emerge. Go back to the unit circle for a minute. We have our radius, 1, centered at the origin. I want to look at the sine values. You can also see this on the table. When I start listing some values of sine, sine is my y values, what happens? I go from y value of 0 to a y value of 1 to a y value 0, to a y value 1. You just go around and around the circle in a counterclockwise direction. The y values go positive, positive, negative, negative when you move from Quadrant 1-4. That pattern, as you work your way around, just repeats and repeats and repeats. We're trying to graph this function. Now, I want the function f of x equals sine of X. At 0, we are 0, and at 2 Pis at four laps, and maybe I'll put 2 Pi on the board here. What happens? I go from 0, at Pi, sine of Pi is also 0. You start filling in the middle here. At Pi over 2 at 90 degrees, I'm 1. Then at 3 Pi over 2 at the bottom of the unit circle, I have negative one. This graph, if you connect the dots, if you graph this thing, you get a very nice symmetric wave function. My hand-drawn picture here isn't doing justice to how pretty this thing is, but maybe you can draw a nicer picture or graph it on the calculator or some online graphing website. If you graph this thing, you'd get a very nice symmetric wave. This wave has a name. It is pretty nice. It's called the sine wave. Or if you want to sound fancy, you can call it the sinusoid, like a dinosaur. The sinusoid. It's a nice graph. I drew the one period of this thing, so if we did one lap around the circle, there's nothing stopping me from going further and just repeating this thing up and down, up and down forever and ever. You can also go counterclockwise. You can go backwards and off it goes. It displays the same symmetry though, so if you just study one cycle, one rotation, one period of this thing from 0 to 2 Pi, you get most of the information you need. You could always unwind or wind more if you need to, but it is, in fact, a function. Give me a number and I'll give you back the sine of that number. Again, what is that? I make a triangle and I described the ratios. That's all this is. If you notice, here's another where notation is going to be a little messy. I put it back in terms of x because it's pretty common to see a function as fx equals x. You have to realize here that this number is often denoted as Theta. So there's nothing stopping you from writing like f of Theta equals sine Theta. These are all dummy variables that we're throwing into the function. I can call it bananas for all I care. It doesn't matter. We can do that as well. For the most confusing one, if you want to write it as y equals sine of x, you have to know from context that y is the output. It is the sine value and x is the angle. This is a little confusing because when we drew the triangle, we are writing it with the x as the x coordinate. Why is that? These are different variables. What we mean, it will be clear in context what's going on. You can take in a circle, you can imagine feeding it any angle, and you can graph the outputs of these functions. With any function, I get to ask you all the same questions that I asked you with any function. In particular, what is its domain? What is its range? What are the x-intercepts or the y-intercepts? What's the end behavior? What does that mean? So that means like as x gets really large, where does the function want to go? As x gets really small, where does the function want to go? Do I have any asymptotes? Clearly from this one, I don't. But you're also allowed to ask that. So if you want to pause the video, take a second, and see if you can fill these in. Think about what these means. Are you ready? The domain, this is a beautiful, nice connected continuous graph, this domain is all reals, it's a nice function, sine of x. Its range is interesting and has a low point, the crest and trough, they are at one and negative one. This function will never get any bigger than one and never get any smaller than negative one, and it hits those values as well, that's at the top of the unit circle and the bottom of the unit circle, it's the maximum y-value and the minimum y-value. So its range is from negative 1-1, and I want square brackets around this thing so that I have these values. X-intercepts, so pretty also known as the zeros of the function. So where does that happen? So we have one at 0, and Pi, and 2Pi, so all the places where y is nice and flat, I guess. So it would be wrong, see if you can figure this out, why is this wrong? If I said x equals 0, Pi, and 2Pi, and walked away? I could clearly here 1, 2, 3, I don't see any more on the right side. But if you look left, you see why? Because this wave goes on, and on, forever, and ever, and ever. So you have another one like negative Pi, that's back here, and negative 2Pi, that's back here, and 3Pi, 4Pi. So there's a couple of different ways to write this. What you can do is you can write plus or minus on each number of 2Pi, if I had room, maybe plus or minus 2Pi dot, dot, dot, and that gets you all the answers. That is one way to do it. The other way to do it, the fancy way to do it, is x equals KPi, for k equal 0, 1, 2, 3, or negative, I should say, for plus or minus. You want to choose a variable to do this, that's fine as well. You can also say k in Z, where Z for Zalon and C for integers. So it has infinitely many. This is the first graph that you've probably seen where you have infinitely many x-intercepts. In terms of y-intercepts, there's only one, right at the origin, 0, 0, just one, so y equals 0. Then the end behavior, this is a wave. So what number does it approach? It turns out it doesn't approach any number. You might say, "Well, does it go to 1 lot?" But yeah, but as soon as you hit 1, you leave. But then does it hit negative 1 a lot? Well, sure. But once you hit negative 1, you'd bounce back up. So what we say in this case here is that there is no specific number that it approaches. In fact, you get this nice way to describe the behavior of a wave called oscillation. So we say the number doesn't exist, there is no such number, DNE, abbreviate does not exists as DNE, this is both true as x goes to infinity and negative infinity, and we say that the wave oscillates, and that's a nice way to describe this behavior. So someone said, "What's the end behavior assigned to me?" I say, "It oscillates between 1 and negative 1," and that's the right way to say it. Same thing is on the backend, and it was the end behavior of x, and the function as x goes to negative affinity oscillates between 1 negative 1, it doesn't approach a single number. Graph of cosine x is very similar, I'm going to go a little quicker through this one, but if you draw the unit circle and you chase the x values on the unit circle, what do you get? So same thing here, and let's graph this thing, we're trying to graph f of x is cosine of x. So what happens at 0? What's the cosine value of 0? You get 1, and period again is 2Pi, so maybe we have to stop at 2Pi, and you have Pi. Well, we could do the intermediate values as well. Pi over 2 and 3 Pi over 2. So follow the cursor here as I walk around the curve. At 0, I get 1, at Pi over 2, I get 0. Then as I get to Pi, I get minus 1, 3Pi over 2 is once again 0, and then 2Pi is 1 as well. So this one here, I'm not going to do it just this. But it's another oscillating curve, and it starts high and goes low and keeps going. Just doing one lap, there's nothing stopping you from doing as many lapses you want. The maximum value is 1 and the minimum value is 1 as well. Now, we're looking at the x-axis due to the symmetry of the circle, that's its best. On a unit circle, you're going to get from minus 1 to 1. So off it goes. Same things on this one. What's the domain of this function or reals? What is the range of this function from minus 1 to 1? Square brackets around minus 1 and 1, because you do hit those values. What are the x intercepts? So the x-intercepts here would be Pi over 2, and so this amount until we see a Pi over 2, 3Pi over 2, and you can get to keep going and see 5Pi over 2. So the pattern is you have odd integers times Pi over 2, and not just on the right side, but also on the backend. So you have plus or minus, plus or minus, plus or minus. The y-intercept, it's a function. So by the vertical line test, you're only going to have one, and that's right at 0, 1. The end behavior, this also has oscillation, so it oscillates between minus 1 and 1. There's no asymptotes on this thing, it's very nice. There's a property of sine and cosine that I think are really important to call out. Once you see the graph and understand the graph, it's pretty obvious. But if you think about [inaudible] some other functions that are like this, it's a little tricky to cook these up. They're certainly out there, they're not the only function, but this is notion of boundedness. So boundedness means that for any value in a domain of a function, that the function has a lower bound and an upper bound. Sine and cosine are great examples of bounded functions because they are bounded below by negative one and above by one. Now, I could put negative 100 and positive 100 sure but the point is they're bounded. As long as you have some upper bound and some lower bound, you are bounded. This fact is going to make these functions even more useful as you go through your mathematical career. The key here is that they have both upper and lower bound. When this happens, you are called bounded. So here's my challenge to you. Can you think of any other function that is bounded besides sine and cosine? It's a little challenging to do. The ones that we've seen like the parabola, or the absolute value function, or the cubic, or exponential logarithm, those are all unbounded function. What's nice about these periodic functions is that you do get nice boundedness in there. So keep this in mind, remember this, we will use this property later. As you go through, ask yourself is this function bounded, or is it not? There's one more thing I want to point out about the graphs of sine and cosine. So let me put both of them on the screen so that you can see here. So I'm going to draw a sine, starts at the origin, goes up, it goes down, it goes up again, keeps going forever and ever. I'm going to draw it one period to the left on the positive x-axis and one period to the right. So we're going 0-2 Pi and I'll say negative two Pi, so two Pi. Not the world's most prettiest graph but hopefully you get the idea. Cosine of x, same thing, I'm going to draw the x, y-axis. I'm going to start at the top, at the point 0,1. I'm going to go nice and smooth in a wave-like form down and then up. Probably should be more symmetric but it's a little hard to draw on a computer and we have this thing. Once again, a 0-2 Pi is one period and then from negative two pi to zero is another period. So I'm joining two periods of these waves. It should be perfectly symmetric. So that's what I want to talk about. As we wrap up this video here, is the symmetry. I want to talk about the symmetry of these two curves. So let's do cosine first. It's a little easier. If you stare at the symmetry of this and again draw a nicer picture than I did. If you realize, you can take the graph and fold it left to right, this is like a butterfly thing. This has y-axis symmetry. So like the parabola, like the absolute value graph, if you're looking on the right, then you can tell me something about the left. This is why mathematicians love symmetry because I don't actually need to look left to tell you what's going on. I don't have to tell you what's going on the other side of the study, it is to have to work and I get all the information I want. This y-axis symmetry is captured by noticing that whatever my positive value is, if I have some positive value Theta and I look at the curve, the opposite is the same as its negative. So you get a nice little observation here that cosine of some value Theta is the same as the cosine of negative Theta. If you know the graph, then this little equality I just wrote, cosine Theta is cosine negative Theta is clear. I can look on the right side and tell you it's not a left side or vice versa, it doesn't matter. When this happens, when you have this thing and it behave like the parabola, we call this an even function. Even function is getting its name, not because of even numbers, whatever, but because x squared has a symmetry as well. X squared now it's getting from the two. When you have y-axis symmetry, you call them even function. What that says is if you plug in the negative value, you get the exact same as the positive value. So negatives don't matter. Now stare at sine of x for a minute. If I fold over the y-axis, I don't quite get the nice symmetric thing I want, but there is symmetry hidden in here. It's just not immediately obvious. Imagine if I put like a little pin at the origin and then rotate this thing around the pin so it has rotational symmetry, you'll land directly on, so you can imagine left hand, right hand, like grabbing the handles, giving them a spin 180 degrees, you get rotational symmetry. Sometimes this is also called origin symmetry because your rotational about the origin. In this case, so what does it mean? So if I have some positive value of Theta and I want to know what is going on at its negative, well, what do you notice? That they're always the same numbers but they differ by a sign. It does not matter where you put in. So if you start writing that out as an equation, you get sine of negative Theta is negative sine of Theta. So for sine values, the negatives pop-out. That is the algebraic way of describing this origin symmetry. When that happens, what something else where negatives pop out, it's behaving like the function execute. In that case, whenever you have a function with this property, where negatives pop out, we call it an odd function. So you got to be careful because I'm using even and odd not the way that we normally do because we normally say like 2, 4, 6, those are even numbers and 3, 5, 7, those are odd numbers. Those are things where we're using that language now to describe symmetry of the graphs of functions. It turns out that sine is an odd function and cosine is an even function. What I would do if I were you is like somewhere in your notebook on the back or some of that, keep track of these observations here, these formulas. We're going to use these and they're really nice because they simplify some things and they save us some work along the way. So we'll use all these identities. We'll do lots of problems in the next video. So keep all these things handy. Great job on this one. I'll see you in the next.