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Sal draws the graph of the sine function to learn about its domain and range. Created by Sal Khan and Monterey Institute for Technology and Education.
Video transcript
We're asked:"What are the domain and range of the sine function?" So to think about that, let's actually draw the sine function out. And what I have here... on the left hand side right over here I've got a unit circle and let me truncate this a little bit I don't need that space over here so let me clear that out. So I have a unit circle on the left hand side right over here and I'm going to use that to figure out the values of sine of theta for a given theta. So on the unit circle this is x and this is y. Or you could even view this as the... Well we could just use X and Y and so for a given theta we can see where the terminal side of the angle intersects the unit circle and the y coordinate of that point is going to be sine of theta. And over here I'm going to graph still y in the vertical axis but I'm going to graph the graph of y = sine of theta and on the horizontal axis, I'm not going to graph x but I'm going to graph theta. We can view theta as independent variable here and theta is going to be in radians. So, we're essentially going to pick a bunch of thetas and then come up with what sine of theta is and then graph it. Let's set up a little bit of a table here. Over here I have theta and over here we're going to figure out what sine of theta is. And we can do a bunch of theta values. We could start at zero. What is sine of theta going to be? When the angle is zero we intersect the unit circle right over there the y coordinate of this is still zero, this is the point (1,0). The y-coordinate is zero, so sine of theta is zero. We can say sine of zero = zero. Now let's try theta is equal to pi/2. I'm just doing the ones that are really easy to figure out. So theta is equal to pi/2, that's the same thing as a 90 degree angle so the terminal side is going to be right along the y-axis. Just like that. Where it intersects the unit circle is right over here, and what point is that? Well, that's the point (0,1). So what is the sine of pi/2? Well, sine of pi/2 is just the y-coordinate right over here is 1. Sine of pi/2 is one. Let's keep going and you might see a little pattern here we're just going more and more around the circle so let's think about what happens when theta is equal to pi. What is the sine of pi? Well we intersect the unit circle right over there that coordinate is (-1,0) Sine is the y-coordinate, so this right over here is sine of pi. Sine of pi is zero. Let's go to 3pi/2. Well now we've gone 3/4 of the way around the circle, we intersect the terminal side of the angle and it intersects the unit circle right over here so based on that what is the sine of 3pi/2? Well, this point right over here is the point (0,-1) the sine of theta is the same thing as the y-coordinate or the y-coordinate is the sine of theta so when theta is 3pi/2, sine of theta is equal to -1. And let's come full circle here. So let's go all the way to theta equalling 2pi. I'll just use the yellow here. What happens when theta is equal to 2pi? Well, then we have gone all the way around the circle and we are back to where we started and the y-coordinate is zero, so sine of 2pi is once again zero. If we were keep going around we're going to start seeing, as we keep incrementing the angle we are going to see the same pattern emerge again. Let's try to graph this. So when theta is equal to zero sine of theta is zero. When theta is equal to pi/2 sine of theta is one. So we'll use the same scale. Sine of theta is equal to one. I'll just make... this is one on this axis and on that axis so we can maybe see a little bit of a parallel here When theta is equal to pi, sine of theta is zero. So we go back right over there. When theta is equal to 3pi/2, so that would be right over here 3pi/2, sine of theta is negative one. So this is negative one over here I'll do the same scale over here, I'll make this negative I'll make this negative one, so sine of theta is negative one and then when theta is 2pi, sine of theta is zero. And so we can connect the dots... You can try other points in between and you get something... you get a graph that looks something like this. My best attempt at drawing it freehand, it looks something like this There's a reason why curves that look like this are called sinusoids. Because they're the graph of the sine function. So just like this... But that's not the entire graph, we could keep going We could go, we could add another pi/2 if you added another pi/2 so if you go to 2pi and then add another pi/2 you could view this as 2 1/2pi or however you want to think about it. Then you're going to go back over here so then you going to get back to sine of theta being equal to one. So you're going to go back to this point right over here and you could keep going, you go another pi/2 you're going to go back to this point and you're going to be over here and so the curve, or the function sine of theta is really defined for any real theta value that you choose. Well what about negatives? Obviously, I agree as you keep increasing theta like this we just keep going around and around the circle and this pattern kind of emerges. But what happens when we go in the negative direction? Well, let's try it out. What happens if we were to take negative pi/2 So let me do that... So negative pi/2, well that's going right over here and so, we intersect the unit circle right over there the y-coordinate is negative 1 so sine of negative pi/2 is negative one and we see that it just continues. So sine of theta is defined for any positive negative, or any theta; positive or negative, non-negative, zero, anything. So it's defined for anything. So, let's go back to the question. I could just keep drawing the function on and on and on. So let's go back to the question. What is the domain of the sine function. And just as a reminder, the domain are all of the inputs over which the function is defined or all of the valid inputs into the function will spit out a valid answer. So what is the domain of the sine function? Well, we already saw. We can put in any theta here. So you could say the domain is all real numbers. Now what about the range? Just as a review the range is, sometimes in more technical math classes it's called the image it's the set of all the values that the function can actually take on. Well, what is that set, what is the range here? What are all the values that y equals sine of theta could actually take on? Well, we see that it keeps going between positive one and then to negative one. And then back to positive one and then to negative one. It takes on all values in between. So you see that sine of theta is always going to be less than or equal to one and it's always going to be greater than or equal to negative one. So you could say that the range of sine of theta is the set of all numbers between negative one and positive one and it includes negative one and one That's why we put brackets here instead of parentheses.