 Welcome back to our lecture series Math 1060 Trigonometry for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In lecture nine and many of the subsequent lectures as well, we're going to talk about graphing the six trigonometric functions. Now, in order to graph a function, whether it's a trigonometric function, algebraic function, or something else, there's a few important ingredients that we need to consider. That is, what's the anatomy of a function graph? What are the things we need to see in the graph to really say that this is the graph of a function? Now, in this video, the things I really want to talk about is the idea of domain range and zeros, which is something you've possibly seen in a previous class about algebra or functions, but we're going to review that right now. So the domain of a function, let's say the function is given as f, this is going to be a set of all values for which f can be evaluated. That is the set of all possible inputs. And similarly, the range of the function f is the set of all values of the form f of c. That is to say c is some number in the domain, we then plug it into the function and we evaluate it. So the range is the set of all possible outputs. And so if we think about that for a moment, we have our x and y axis, maybe something like this. We have x right here and y right here. And suppose we have a function that's doing something like this. So this is our function. This is just some function f right here. So the domain is going to be the set of all points on the x-axis for which there's something that actually is associated to it. It's like, here's a point in the domain, here's a point in the domain. All of these functions are defined right here. Excuse me, the functions defined for all these different points. And for the most part, when it comes to least trigonometry, the only time we have to worry about being outside the domain is when we divide by zero. That's not a problem for sine and cosine, but that will be a problem for the four other trigonometric functions. So division by zero is the main issue we get when it comes to domain. Range, on the other hand, can be often be restricted based upon how high and how low the function can go. Sort of like the limbo line right there. If you look at this function right here, we come down here. But we can come up to here, coming down here, all the way up to here, and then down again. We see that these are all these values in the domain. If we look at the y coordinates here, the y coordinates include all these points right here. Then going back down, we get all the points down here. Then we go back all the way up to the points up here, and then we go down again. So the range is going to be all of the y coordinates that actually appear on the graph. Again, this is something we've probably seen in a previous algebra course. The next thing we want to talk about are the zeros or the so-called x intercepts of the graph. Where does the graph cross the x-axis? So the zeros are exactly those numbers c, such that f of c is equal to zero. Where are these x intercepts? So looking at the graph, we can see there's an x intercept here, and here, and here. Without more of the picture, that's the only thing we could suggest at this moment. So these are going to be the x intercepts of this graph right here. So if we switch over, if we switch our focus as trigonometric functions, what can we say about them? What about sine and cosine? What are their domains? What are their ranges? What are their zeros? So to help us with this, we're going to think about the unit circle diagram. That is, we have the x and y-axis from before. If we think of the typical unit circle, that's a respectable looking circle. It's close enough, right? It looks circular in shape. What are the possible angles that can go inside of sine? So when you think of the domain, we want the possible input. The input of sine is going to be the angles. What angles is sine defined for? So if we think about these possible terminal sides of an angle, here's theta. Well, theta would be defined for that angle. Sine would be defined for that one, and this one, and this one, and this one, and this one. It turns out that any angle you choose, sine is going to be defined, because what is sine after all? Sine is the y-coordinate of these points on the unit circle. So sine theta is just the y-coordinate. No matter where you are on the unit circle, the y-coordinate is defined, so sine will be defined as well. So this tells us that the domain of sine is going to be all real numbers. So it's domain. So we write that out. The domain of sine is going to equal all real numbers. In interval notation, we write that from negative infinity to infinity. So when we graph the sine function, its range will be all real numbers. This is also true for cosine for the same reason. It doesn't matter which angle you choose. The x-coordinate is defined on the unit circle. Therefore, that point will be defined. So the domain of sine and cosine is going to be all real numbers. Range is a little bit different, though. If we think of this angle right here, the y-coordinate at this point at zero degrees, the y-coordinate is going to be zero. So sine can't equal zero. As we go increase the angle in the first quadrant, we can get all the way up to a 90-degree angle, in which case the point is going to be zero, one. So sine can get as big as one, but in the second quadrant, as you pick these other angles right here, you're going to get smaller, smaller y-coordinates on the unit circle coming down to the point negative 1, 0. So we're back down to zero. If we continue down, that is picking smaller and smaller angles, I should excuse me, larger, larger angles, this is going to go smaller and smaller y-coordinates, till we get down to this point here, which we're going to get zero, negative 1, which then in the fourth quadrant, the angles as they get bigger, bigger, bigger, we'll get larger and larger y-coordinates going back to 1, 0. And then you'll notice that the graph is just going to, well, I should say the sine ratios will repeat themselves over and over and over and over and over again after that. So when it comes down to sine, we see that sine, it can be zero up to 1, down to zero, back to negative 1, and does it all over and over again. Of course, we get things in between it, like sine can equal one half if we take 30 degrees, but it turns out sine never gets bigger than 1. The biggest sine could ever be, the biggest y-coordinate on the unit circle is at the very tip top of the circle. The biggest sine could be is 1. The very smallest that sine could be is at the very bottom of the unit circle, which is going to be negative 1. And so for that reason, the range of sine is going to be negative 1 to 1, and this is the interval notation here, y can, can equal negative 1 on the unit circle, so so can sine, and then y could equal 1 on the unit circle. Therefore, sine, that's the biggest value it could be. So sine is bounded above by 1, and it's bounded below by negative 1. So if we make that recording right here, we see that the range, the range of sine is going to equal negative 1 to 1. Now, this same statement is going to be true for cosine as well. If we think of the angle, as the angle goes around, you have zero 1, you go up to the top of the unit circle, you get zero, excuse me, you get one zero first, then you get zero one second, then you come down here, you're going to get negative one zero, come all the way down here, you're going to get zero negative one, and then back to one zero. In this situation, now if we focus on the x-coordinates, the x-coordinate starts at one, it shrinks to zero, then it continues to shrink to negative one, then the x-coordinate is going to grow, grow, grow, grow until we get to zero, then it will grow, grow, grow, grow, grow until we get back to one, and so we see that cosine, the x-coordinate will also be bounded by these two numbers just for different reasons, right? The biggest x-coordinate could be is on the right-hand side at one, the smallest x-coordinate could be is negative one on the left-hand side of the unit circle. So we see that the range of cosine is likewise going to equal negative 1 to 1. So domain and range of sine and cosine are actually the same thing, sine and cosine have the same domain, sine and cosine have the same range, all right? The one important difference though is going to be the zeros. Where is sine equal to zero? Where is cosine equal to zero? So putting these points back on the screen here, they seem to be important to us here. If we look at sine for a moment, like so, where's sine, where's the sine coordinate, excuse me, when's the y-coordinate equal to zero? Well that's going to happen when you're on the y-axis right here. When you're on the y-axis, that means start again in 3, 2, 1. Where is the y-coordinate equal to zero? Well that's going to happen on the x-axis right here. Notice that when you're on the x-axis, that's exactly those points whose y-coordinate is zero. These are going to be the places right here where sine of theta is equal to zero. It happens here and it happens here. This tells us that sine theta equals zero exactly when theta equals zero or pi radians, 90 degrees, same thing. But this is of course only when you go around the unit circle once. When you go around the unit circle again, it's going to repeat itself. So if you have zero, you have pi, you're also going to get two pi because that's the same thing as zero. Then you're going to get three pi, that's the same thing as pi. Then you're going to get four pi, that's coterminal to zero. You're going to get five pi, which is coterminal to pi. And that's also that if you go the other direction as well, if you go clockwise, negative pi, negative two pi, negative three pi, all of these coterminal angles will also give you zeros as well. The zeros are going to be showing up over and over and over again for sine. So when it comes to sine, it turns out that the zeros happen at multiples of pi, integer multiples of pi. So zero pi, one pi, two pi, three pi, negative seven pi, that's where the zeros of sine occur. So sine theta equals zero when theta is a multiple of pi. For cosine, where is this going to happen? Well, if you're looking for a cosine, which measures the x-coordinate, the x-coordinate is going to be zero when you're on the y-axis, like we see right here, x is zero and x is zero, just like so. And so for cosine, cosine will be zero at pi halves, 90 degrees, at three pi halves, 270 degrees, and then you're going to repeat itself. You go up here, this would be five pi halves, then you're going to get seven pi halves, then nine pi halves, then 11 pi halves, and you just go over and over and over again. And of course, if you go clockwise, you're going to get negative pi halves, negative three pi halves, negative five pi halves, negative, where are we at? Negative seven pi halves, you're going to get the same things over and over again, but in different locations, of course. So for cosine, for cosine, we should mention that cosine will be zero when theta is either pi halves or three pi halves. But also, if you add any multiple of two pi to that, you're going to get another zero as well. So the zeros of cosine are going to be pi halves plus any multiple of pi. When discussing the graphs of sine and cosine, it's often important to talk about their amplitude. This will be especially true when we start modifying the sine and cosine waves in future lectures. So in general, for a function of y equals f of x, let capital M be the largest value in its range, and let little m be the smallest value in its range. So M is the biggest, little m is the smallest. Then we say the amplitude of a function is going to be the difference between the large value with the small value, and we take half of that. So basically it tells you, on average, how far away from the middle can you go? And this is the idea of amplitude, and we're going to see this later on with our trigonometric functions. The amplitude tells you how far above the x-axis you go and how below the x-axis you go. So the amplitude is a way of measuring the size of the range of a function. Now sine of theta is always bounded above by one. That is sine is less than or equal to one, but it's bound to below by negative one. Sine is greater than or equal to negative one, and so this gives us the capital M and the lower case M. So the amplitude for sine is going to be one-half, the absolute value of one minus negative one, like so. So the reason you take this difference, this tells you how big the range is going to be. So you're going to get two right there. The absolute value of course is two, so you get two over two, which gives you one. So for your typical sine function, it's going to go one above the x-axis and one below the x-axis, and it does that about equally and likely. Cosine does the same thing. Cosine is bounded above by one. It's bounded below by negative one, so the amplitude for cosine is likewise going to equal one. We won't go into as much detail with tangent, secant, cotangent, and cosecant as we did with sine and cosine, but let's briefly talk about their range domain and zeros as well. So for the domain of tangent and secant, the thing to remember is that tangent is the same thing as sine theta over cosine theta, and so it's a fraction. Sine and cosine are defined for any angles, but fractions can be undefined if the denominator is equal to zero. So when is tangent undefined? Well, that happens when cosine is equal to zero. When is cosine equal to zero? Well, that happens, as we saw earlier, when theta equals pi-halves plus a multiple of pi. So the domain of tangent is going to be all numbers except for these ones right here. At this place, you're going to get a vertical asymptote. Secant, on the other hand, secant is equal to one over cosine, so it has the same problem. The numerator is always one, but the denominator is going to be cosine, and so if cosine is zero, that means secant will be undefined. So tangent and secant will be undefined when cosine is equal to zero. In terms of their x-intercepts, though, when is tangent equal to zero? Well, you have a fraction. The only way a fraction can equal zero is if its numerator is equal to zero. So tangent is equal to zero exactly when sine is equal to zero, and as we observed earlier, sine is equal to zero when theta equals a multiple of pi. Now, on the other hand, can secant ever equal zero? Well, a fraction, like I said, can only equal zero if the numerator is equal to zero, but the numerator here is one. That can't equal zero. So it turns out that secant can never equal zero as it's the reciprocal function to cosine. And so if we say a little bit about the range right here, it turns out that tangent can actually be, can equal all real numbers. We'll give a little bit more detail about that when we talk about the graphs of tangent. Secant, it can be most numbers, can get arbitrarily big. That is, you can, secant can go off towards infinity or negative infinity, but you can't get small values of secant. That is to say secant, secant of theta, it is always greater than or equal to, well, I should say its absolute value is always greater than or equal to one. Secant can be positive or negative. It's always going to be greater than one in terms of its absolute value. You can't get stuff that's close to the x axis and we'll talk about that a little bit more detail later. Cotangent and cosecant basically do the same thing. Cotangent, of course, is the same thing as cosine theta over sine theta. And so when is, when is cotangent undefined? It'll be undefined when sine is equal to zero and sine is equal to zero at multiples of pi. Cosecant, same thing. Cosecant is one over sine theta. So when sine is equal to zero, cosecant is undefined. In terms of their zeros, right? Cotangent is equal to zero when cosine is equal to zero. That happens at pi halves plus a multiple of pi. Cosecant can never equal zero because the numerator is always going to be a one there. In terms of their range, same basic thing as before, we'll go over more details about this later. Cotangent can equal all real numbers and cosecant, if you take its absolute value, its absolute value is always great and equal to one. That is, cosecant is either larger than equal to one or less than equal to negative one. So the opposite of what sine is doing and we'll get into those details later on.