– [Narrator] 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 I can, let me

truncate this a little bit. I don’t need that space right there, so let me clear that out. So I have a unit circle

on the left hand side right over here, and I’m

gonna 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 do this, as the, well, we can just use X or Y,

and so for a given theta we can see where that angle

entered to the terminal side of the angle intersects the

unit circle, and then those, 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 gonna graph the graph

of Y is equal to sine of theta. Y is equal to sine of theta,

and on the horizontal axis I’m not gonna graph X,

but I’m gonna graph theta. You can do theta as the

independent variable here, and it’s gonna be 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. So let’s set up a little

bit of a table here. Let’s set up a little bit of a table. And so, over here I have theta, and over here we’re going to figure out what sine of theta is, and we could do a bunch of theta values. We could start, we could start,

let’s say we start at zero. Let’s say we start at

theta is equal to zero. What is sine of theta gonna be? Well, 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, this is

the point one comma zero. The Y coordinate is zero,

so sine of theta is zero. We could say sine of

zero is equal to zero. Sine of zero is equal to zero. Now let’s try theta is

equal to pi over two. Theta is equal to pi over two. I’m just doing the ones that

are really easy to figure out. So if theta is equal to pi over two, 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, and where it

intersects, where it intersects the unit circle is right over

here, and what point is that? Well that’s the point zero comma one. So what is sine of pi over two? Well sine of pi over two is just the Y coordinate right over here. It is one. Sine of pi over two 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’s, what, what happens when theta is equal to pi. When theta is equal to

pi, what is sine of pi? Well, we intersect the unit

circle right over there. That coordinate is negative one, zero. Sine is the Y coordinate, so this right over here is sine of pi. Sine of pi is zero. Let’s go to three pi over two. Three pi over two, well now

we’ve gone three quarters of the way around, around the circle. We intersect the terminal

side of the angle intersects the unit circle right over here, and so based on that what is

sine of three pi over two? Well, this point right over

here is the point negative, we gotta be careful, is

zero, is zero negative one. The sine of theta is the

same thing as a Y coordinate if the Y coordinate is

the sine of theta, so theta, when theta’s pi over two sine of theta, or when theta’s three pi over two sine of theta is equal to negative one. And let’s come full circle. Let’s come full circle here. So let’s go all the way

to theta equaling two pi. Let me do a color, hey, I’ll

just use the yellow here. What happens when theta is equal to two pi? Well then we’ve gone all

the way around the circle, and we are back to where we started, and the Y coordinate is zero, so sine of two pi is once again zero, and if we were to keep going around, we’re gonna start seeing as we

keep incrementing the angle, we’re gonna start seeing the

same pattern emerge again. Well, let’s try to graph this. So when theta is equal to

zero, sine of theta is zero. When theta is equal to pi

over two, when theta is equal to pi over two, pi over

two, sine of theta is one. So, we’ll use the same scale. So sine of theta, sine

of theta is equal to one. This is, I’ll just make this, 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 when theta is equal to

pi, sine of theta is zero. So we go back right over there. When theta is equal to three pi over two, so that would be right over

here, three pi over two, 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, let me make down a little bit, I’ll make this negative one, and so, sine of theta is negative one. And then, when theta is two

pi, sine of theta is zero. So when theta is two pi, two

pi, sine of theta is zero. And so we can connect the dots. You could try other points in between and you get something, you get a graph that looks something like this. It looks something like this. My best attempt at drawing it freehand. It looks something, something like this. There’s a reason why curves

that look like this are called sinusoids, because they’re

the graph of a sine function. So this like, just like this, but that’s not the entire graph. We could keep going. We could go, we could

add another pi over two. If you added another pi over

two, so if you go to two pi, and then you add another pi over two. So you could view this

as two and a half pi, or however you wanna think about it, then you’re gonna go back over here. So then you’re gonna get back to sine of theta being equal to one. So you’re gonna go back to

this point right over here, and you could keep going. You go another pi over

two, you’re gonna go back to this point, and you’re

gonna be over here, and so the curve, the curve

or the function sine of theta is really defined for any theta value, any real theta value that you choose. So any theta value. Well, what about negatives? I mean 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, if we were to take negative pi over two? So let me do that. So negative pi over two, well,

that’s going right over here, and so we intersect the unit

circle right over there. The Y coordinate is negative one. So sine of negative pi

over two is negative one, and we see that it just continues. 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. So I could just keep drawing

this function on and on and on. So let’s go back to the question. What is the domain, what is the domain? What is the domain of

sine 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 that the function will actually

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, the domain is all, all real numbers, all

real, all real numbers. Now, what about the range? What about the range? Well just as a review,

the range is sometimes in more technical math

classes 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 is all the values that Y equals sine of theta could actually take on? Well, we see that it keeps

going between positive one, it keeps going between positive

one and then to negative one and then back to positive

one and then negative one. It takes on all the values in between. So you see that sine of

theta, 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, and that’s why we put brackets

here instead of parentheses.

## 14 Comments

## Jonathan McDonnell

Thank you so much! This topic came up in a book i was reading and when i asked others for help they just said that it was too complicated. Your video was extremely helpful, thorough, and simple. Thank you very much!!

## taylor mekdalan

god of education i thank you!

## ALI HAIDER

thanks a lots

## Mohammad Bilal

I hope Sal you would do the videos on domain and range of all the trigonometric functions and their inverses! I find a little trouble while analyzing the domain and ranges of some complex trig functions…

– via YTPak(.com)

## Shujat Ali Khan

Someone told me that Domain of sin x is from 0 to 2π. How's that possible?

## Winnie Dai

at the end of the day, only sal can help me.

## Winnie Dai

at the end of the day, only sal can help me.

## G'S Up 34

What the fuck is this shit

## Bajodah

What is this? 😔

## Weikang Hu

The only thing confusing me is 3pi/2, how is that -1? Can someone help plz:)

## ryehaaan

ive learned this when i was in high school but completely forgot. so i need to revise for my uni XD

## verifiedmartian

Watching on x2 speed 10m before a quiz. Help me.

## Caleb Rossouw

Can you stop using π please 😂😂

## Martin Bartsch

mr. weber brought me here