Inputs into the function that the function will actually Sine of the sine function? And just as a reminder, theĭomain are all of the inputs over which the function is defined, or all of the valid What is the domain, what is the domain? What is the domain of So sine of theta is definedįor any positive, negative, or any theta, positive or negative, non negative, zero, anything. Over two is negative one, and we see that it just continues. That's going right over here, and so we intersect the unitĬircle right over there. Take, if we were to take negative pi over two? So let me do that. In the negative direction? Well, let's try it out. Well, what about negatives? I mean obviously I agree,Īs you keep increasing theta like this we just keep goingĪround and around the circle and this pattern kind of emerges, but what happens when we go Or the function sine of theta is really defined for any theta value, any real theta value that you choose. Gonna be over here, and so the curve, the curve Two, you're gonna go back to this point, and you're This point right over here, and you could keep going. So then you're gonna get back to sine of theta being equal to one. So you could view thisĪs two and a half pi, or however you wanna think about it, then you're gonna go back over here. Two, so if you go to two pi, and then you add another pi over two. We could go, we couldĪdd another pi over two. So this like, just like this, but that's not the entire graph. That look like this are called sinusoids, because they're You could try other points in between and you get something, you get a graph that looks something like this. I'll make, let me make down a little bit, I'll make this negative one, and so, sine of theta is negative one. Here, three pi over two, sine of theta is negative one. When theta is equal to three pi over two, so that would be right over So we can maybe see a littleīit of a parallel here. This is, I'll just make this, this is one on thisĪxis, and on that axis. Over two, when theta is equal to pi over two, pi over Keep incrementing the angle, we're gonna start seeing the 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 What happens when theta is equal to two pi? Well then we've gone all 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. Same thing as a Y coordinate if the Y coordinate is Here is the point negative, we gotta be careful, is Sine of three pi over two? Well, this point right over Side of the angle intersects the unit circle right over here, and so based on that what is We've gone three quarters of the way around, around the circle. Sine is the Y coordinate, so this right over here is sine of pi. Pi, what is sine of pi? Well, we intersect the unitĬircle right over there. So let's think about what's, what, what happens when theta is equal to pi. So what is sine of pi over two? Well sine of pi over two is just the Y coordinate right over here. Here, and what point is that? Well that's the point zero comma one. Intersects, where it intersects the unit circle is right over To be right along the Y axis just like that, and where it So if theta is equal to pi over two, that's the same thingĪs a 90 degree angle.
I'm just doing the ones thatĪre really easy to figure out. Now let's try theta isĮqual to pi over two. What is sine of theta gonna be? Well, when the angle is zero we intersect the unitĬircle right over there. 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. So let's set up a littleīit of a table here. What sine of theta is, and then graph it. To pick a bunch of thetas and then come up with
Independent variable here, and it's gonna be theta is Y is equal to sine of theta,Īnd on the horizontal axis I'm not gonna graph X,īut I'm gonna graph theta. Still Y in the vertical axis, but I'm gonna graph the graph Unit circle, and then those, the Y coordinate of that point is going to be sine of theta. This is X, and this is Y, or you could even do this, as the, well, we can just use X or Y,Īnd so for a given theta we can see where that angleĮntered to the terminal side of the angle intersects the Gonna use that to figure out the values of sine of On the left hand side right over here, and I'm I don't need that space right there, so let me clear that out.
Over here I've got a unit circle, and I can, let me Let's actually draw the sine function out,Īnd what I have here, on the left hand side right What are the domain and range of the sine function? So to think about that,
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