All right 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 very similar thing, but basically 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, has radius 1, and the idea is that if we draw a triangle with the vertex of the origin and whose hypotenuse is the radiant, it forms an x and a y as its adjacent and opposite sides an we create a little angle theta. And we use the fact that, well, y, well, let's do it this way, the sin(theta) was y and cos(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 1. So you can imagine if you're trying to measure some buildings, you do some architecture, you're doing whatever modeling you want to do, well, your triangle may not be of size 1. And so you'll start to see word problems where you draw a triangle and the given information is just larger. So for example, a 3, 4, 5 triangle, okay? 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 this x and this y, so at the vertex of the triangle, so x = 3 and y = 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 5. 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 this point (3, 4), well, it's not the unit circle. The radius is clearly 5, so you can find that 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 + y squared all under the square root. So r = square root of x squared + y squared and in that general case, in the general case, you may not get 1, so may not equal 1, and that is okay. Not every triangle lives inside 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 SOH, sine is opposite over hypotenuse and cosine is adjacent over hypotenuse. How does this change our formula? So sin(theta), so how does this change our formula? So sin(theta) which is normally opposite over hypotenuse, becomes opp/hyp. Well, in our triangle here, that's not the unit circle, since our hypotenuse is not 1 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 cos(theta) will become our adj/hyp, and that becomes x/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 sort of looking at the triangle and grabbing the x and y value that we had before in previous videos, sin(theta) is y, cos(theta) is x, 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 1, 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 and 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 this is all about you comparing math to cooking. Yes, there's a like a billion recipes out there, but you always have your one thing that you go through, 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 where I have Hypotenuse r, I have my y value in my x value for my adjacent opposite sides and again, you're imagining it in some sort of unit circle and some sort of circle. Maybe it's unit, maybe it's not whatever you want. In that case, we have that x of course is 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. You see it this way as well, it's good to get used to doing both ways again. Again if you want a picture of the unit circle, just replace r with one, that's not a problem. And then there's one identity that you have to know. You say no, I don't want to have to understand and learn more identities. The good news is this is nothing new. You're dealing with a right triangle. What is the one identity that like I don't know when they start teaching it, it's pretty early. It's pretty early like middle school, elementary school, it's like 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 Shmagarian theorem. It's the Pythagorean Theorem. Yeah, of course, so the sum of the sides squared hypotenuse squared, so x squared plus y squared is equal to r squared in our case. This is the Pythagorean Theorem. Okay, I would like to trust the elementary education system that people have seen this before. I hope this is not something new like well, I've never seen that before. Okay, so x squared plus y squared is the radius. Sometimes it seems like a, b and c, whatever the variables are, they're all dummy variables, who cares? But the sides squared, sum them up is equal to the hypotenuse squared. In our case we will work with right triangles, so I always want to replace it 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 allowed over here so we're going to bring the two into both the r and the cosine hits both pieces and you get r squared cosine squared theta plus r squared sine squared theta, so same thing. Bring the two in on the r and the sine, that's all equal to r squared. And then divide every single piece by r squared. You can certainly do that whatever you do on 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 1. Notice this identity is if I had done with the unit circle, I would get the same thing cosine squared plus sign squared this one. This is the fundamental I want to call it the fundamental Pythagorean identity. This is like 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 1, 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 on parentheses squared but they don't do that for whatever reason. They write it as cosine squared of theta. They put the 2 on the cosine because they don't want to confuse anybody. Do you mean like am I squaring the theta fist 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 and often often often when you don't have much going on as if it's just theta, you don't write the parentheses. This becomes cosine squared of theta. The parentheses are sort of 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 and 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 do, let's find cosine of theta if sine theta is 3/5 and theta is in quadrant two. All right, so what does that mean? Again, when you get these all this information, I always think it's best to draw pictures trying to visualize if you're visual learner see what's going on here. So theta is some theta in quadrant two and we go counterclockwise when we label are quadrants. So theta is somewhere back here. So we have some ray that ends up in quadrant Do and 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 sort of circle I'm on, so why don't we can? Well, I guess they kind of do, they tell me the sine of theta is through 5, so remember SOH CAH TOA. SOH CAH TOA, SOH CAH TOA, I'll it all out, sine is opposite over hypotenuse. So our hypotenuse is drawn right angle, over here this are opposite side. There's good old little triangle here is going to be 3. All right, hypotenuse is 5 and I want the missing base of a side, so maybe I'll come over and draw a new larger triangle. So don't try to squeeze it into my small picture here, so we have some other angle theta, the thing that the reference angle that we want. If right triangle, they're telling me that sine of theta surface, I don't know what theta is, I 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 3 and the hypotenuse is 5. Given two sides of a right triangle, you can always solve for the other side, so you do 5 squared. You can call it the x, if you want, 5 squared is x squared + 3 squared. You may recognize these numbers already is a 3, 4, 5 triangle. Let's go through the algebra, so you get 25 is x squared + 9, subtract 9 both sides, x squared is 16 and of course, x is 4. The algebraist I'd be yelling at me when I take a square root, you're supposed to plus or minus. This is a 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 in the circle, the radius is 5, so we know x is 4. So they're asking for cosine of theta, cosine of theta. Remember SOH CAH TOA, SOH CAH TOA, so cosine of theta would be adjacent over hypotenuse. The adjacent side, language 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 gotta remember, I'm using the reference angle. I'm using the helper angle to find this piece and 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 two, whenever this x value is over here, I am negative. I am very negative, so the first piece of information sine of theta surface will give you what the ratio is. Just as you can use Pythagorean theorem to find the missing third side. And the second piece of information will tell you if it's plus or minus quadrant to the x value is negative. So this thing final answer here, the cosine value that I want, because I'm in quadrant two is negative four-fifths. So final answer there. We start to treat the sine or cosine as a functional unit circle. A pattern starts to emerge, so go back to the unit circle for a minute. We have our radius one center of the origin and I want to look at the sign values, but 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 1 to y value 0 to a y value 1 and you just go around and around the circle in a counter clockwise direction. The y values go positive, positive and negative, negative when you move from quadrant one to four? That pattern, as you work your way around just repeats and repeats and repeats. So we're trying to graph this functions and I want the function f of x equals sine of x. So at 0, we are 0, and at 2 pi is in full laps, maybe I put 2 pi on the board here. What happens? I go from 0 at pi, sine of pi is also 0. You can start filling in the middle here, so at pi over 2 at 90 degrees, I'm 1. And then at like 3 pi over 2 at the bottom of the unit circle, I'm negative 1. And so this graph if you connect the dots, you graph this thing, you get a very nice symmetric wave function. My hand drawn picture here isn't doing just as how pretty this thing is, but maybe you can draw a nicer picture or graph, another calculator or some online graphing website. But if you graph this thing, you get a very nice symmetric wave. This wave has a name, it's pretty nice, it's called the sine wave or if you want to sound fancy, you can call it the sine of Soid, sounds like a dinosaur. The sinusoid, so 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 can always unwind or wind more if you if you need to, but it is in fact a function. You 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 describe the ratios, that's all this is. So if you notice, I put it in terms of, here's another where notation is going to get a little messy, I put it back in terms of x, because it's pretty common to see a function as f of x equals x. We 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. So we can do that as well. And of course, 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. And this is a little confusing, because when we drew the triangle, we were writing it will be x is the x coordinate and y is that, so these are different variables. And what we mean, it will be clear in context what's going on. So you can take the inner circle. You can imagine feeding it any angle and you can graph the outputs of these functions. And with any function, I get to ask you all the same questions that I ask you with any function. In particular, what is its domain, what is its range? What are the x intercepts? What are the y intercepts? What's the end behavior? What does that mean? So that means as x gets really large, where does the function want to go? And 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, pause the video, take a second, and see if you can fill these in. Think about what this means. The domain, this is a beautiful nice connected continuous graph. Its domains is all reals. It's a nice function, sine of x. Its range, now, this range is interesting. It has a low point, the crest, and trough, they are at 1 and -1. This function will never get any bigger than 1 and never getting smaller than -1, and it hits those values as well. That's at the top of the inner circle and the bottom of the inner circle, with the maximum y value and the minimum y value. So its range is from -1 to 1, and I want square brackets around this thing, so that I have these values. x intercepts, also known as the 0s of the function. So where does that happen? So we have one at 0 and pi and 2 pi, 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 = 0, pi, and 2 pi, and walked away, right? Clearly they're here, 1, 2, 3, I don't see any more on the right side, but if you look left, you see y, because this wave goes on and on forever and ever and ever. So you have another one like negative pi, it's back here, and negative 2 pi, it's back here, and 3 pi, 4 pi. So there's a couple different ways to write this. What you can do is you can write plus or minus on each number, 2 pi, if I had room we could do do plus or minus 3 pi, dot dot dot, and that gets you all the answers. That's one way to do it. The other way to do it, the fancy way to do it is x = k pi, for k = 0, 1, 2, 3, or negative I should say, for plus or minus. If you want to use like a variable to do this, that's fine as well. You can also say k and z, or z 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, it's right at the origin, 0, 0, just one. So y equals 0, and then the end behavior, this is a wave. So what number does it approach? Turns out it doesn't approach any number. You might say, well, doesn't it go to 1 a lot? Yeah, but as soon as you hit 1 you leave. But doesn't it hit -1 a lot? Well, sure, but once you hit -1 you bounce back up. So what we say in this case here is that there is no specific number that it approaches, and 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 exist as DNE. This is both true as x goes to infinity and negative infinity, and we say that the wave oscillates. 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 -1. And that's the right way to say it. Same thing is on the back end. What's the end behavior of the function as x goes to negative infinity? Oscillates between 1 and -1. It doesn't approach a single number. Graph of cosine x is very similar. I want to go little quicker through this one, but if you draw the unit circle an you chase the x values on the unit circle, what do you get? So same thing here, 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 0? You get 1. And period again is 2pi, so maybe I'll just draw 2pi, and you have pi. So at pi, well, we could do the intermediate values as well, pi/2 and 3pi/2. So follow the cursor here as I walk around the curve, at 0, I get 1, pi/2, I get 0. And then as I get to pi, I get -1, 3pi/2 is once again 0, and then 2pi is 1 as well. So this one here, I'm not going to do it justice, but it's another oscillating curve. And it starts high and goes low and keeps going and just doing one lap. There's nothing stopping you from doing as many laps as you want. The maximum value is 1 and the minimum value is 1 as well. Now we're looking at the x axis to do the symmetry in the circle that says best on the unit circle, you're going to get from -1 to 1. So off it goes, same sort of things on this one. What's the domain of this function? All reals. What is the range of this function? From -1 to 1, square brackets around -1 and 1 because you do hit those values. What are the x intercepts? So the x intercepts here would be pi/2. We start listing them out until we see a pattern pi/2, 3pi/2, and you could get if you keep going, you'll see 5pi/2. So the patterns, you have odd, powers are odd integers, times pi/2, and not just on the side, but also on the back end. 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 1, and that's right at 0, 1. And the end behavior, this also has oscillation, so it oscillates between -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. And once you see the graph, understand the graph, it's pretty obvious, but if you think about it, if I said do you name some other functions that are like this? It's a little tricky to cook these up there, they're certainly out there, 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 upper bound. Sine and cosine are great examples of bounded functions because they are bounded below by -1 and above by 1. Now, I could have put like a negative 100 and a 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. And this fact is going to make these functions even more useful as you go through your mathematical career. Okay, when this happens, when you have both, the key here is that they have both upper and lower bound. When this happens, you're 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 boundaries in there, so keep this in mind. Remember this, we will use this property later, and 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 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 to 2pi, and also, -2pi to 2pi. So not the world's most prettiest graph, but hopefully you get the idea. And cosine of x, same thing, I'm going to draw the XY 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, it's a little hard to draw on the computer. And we have this sort of And once again, at zero to 2pi is one period and then from -2pi to 0 is another period. So I'm trying two periods of these at least waves. It should be perfectly symmetric, okay? So that's what I want to talk about and as you wrap up this video here is the symmetry. So we'll talk about the symmetry of these two curves. So let's do cosine first, I think it's a little easier. If you stare at the symmetry of this and again draw nicer picture than I did. If you realize you can take the graph and fold it left to right. Folder left to right this is like a butterfly kind of thing. This has y-axis symmetry. Select 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 just do half the work and I get all the information I want. This y-axis symmetry is captured by noticing that whatever my positive value is, I have some positive value theta and I look at the curve, the cosine of the value it's exactly the same as, the output 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 show cosine theta is supposed to be- theta is clear. I can look on the right side and tell you it's going to left side or vice versa. It doesn't matter when this happens when you have this sort of thin when you behave like the parabola, we call this an even-function. Even function is getting his name not because of even numbers whatever but because 2 squared has a symmetry as well and x squared now is getting from the 2. When you have y-axis symmetry, you're called an even function, but that says if you plug in the negative value, you get the exact same as the positive value. So negatives don't matter. Now stare sine of extra minute. If I fold over the y-axis, I don't quite get the nice symmetric I think I want. But there is symmetry hidden in here. It's just not immediately obvious. Just Imagine if I put like a little pin at the origin and then will rotate this thing around the pin. So it has like rotational symmetry you'll land directly on slick imagine left hand, right hand like grabbing 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 is going on at its negative. Well, what do you notice that they're always the same numbers, but they differed by a sine. 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 were like negatives kind of pop-out. If b is behaving like the function x cubed. In that case, whenever you have a function that with this property or negative pop-out, we call it an odd function. So you gotta be careful because I'm using even and odd, not the way that we normally do. because you normally say like 2, 4, 6 those are even numbers and 3, 5, 7 those are odd numbers. Those are things 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. So what I would do if I were you is somewhere in your notebook on the back or something that keep track of these little things 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. All right, so we'll use all these identities will do lots of problems in the next video so keep all these things handy. Great job on this one. I'll see you next.
