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Well, we know one revolution of a circle is 2 pi radians. And how many degrees is that? If I do one revolution around a circle, well, we know that that's 360. I can either write it with a little degree symbol right like that, or I could write it just like that. And this is really enough information for us to think abou... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
I can either write it with a little degree symbol right like that, or I could write it just like that. And this is really enough information for us to think about how to convert between radians and degrees. If we want to simplify this a little bit, we can divide both sides by 2, and you could have pi radians are equal ... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
And that's also 180 degrees. And if you want to really think about, well, how many degrees are there per radian, you can divide both sides of this by pi. So if you divide both sides of this by pi, you get one radian. I have to go from plural to singular. One radian is equal to 180 over pi degrees. So all I did is I div... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
I have to go from plural to singular. One radian is equal to 180 over pi degrees. So all I did is I divided both sides by pi. And if you wanted to figure out how many radians are there per degree, you could divide both sides by 180. So you'd get pi over 180 radians is equal to 1 degree. So now I think we are ready to s... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
And if you wanted to figure out how many radians are there per degree, you could divide both sides by 180. So you'd get pi over 180 radians is equal to 1 degree. So now I think we are ready to start converting. So let's convert 30 degrees to radians. So let's think about it. So I'm going to write it out. And actually, ... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
So let's convert 30 degrees to radians. So let's think about it. So I'm going to write it out. And actually, this might remind you of kind of unit analysis that you might do when you first did unit conversion, but it also works out here. So if I were to write 30 degrees, and this is how my brain likes to work with it, ... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
And actually, this might remind you of kind of unit analysis that you might do when you first did unit conversion, but it also works out here. So if I were to write 30 degrees, and this is how my brain likes to work with it, I like to write out the word degrees. And then I say, well, I want to convert to radians. So I ... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
So I really want to figure out how many radians are there per degree. So let me write this down. I want to figure out how many radians do we have per degree. And I haven't filled out how many that is, but we see just the units will cancel out. If we have degrees times radians per degree, the degrees will cancel out and... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
And I haven't filled out how many that is, but we see just the units will cancel out. If we have degrees times radians per degree, the degrees will cancel out and I'll be just left with radians. If I multiply the number of degrees I have times the number of radians per degree, we're going to get radians. And hopefully ... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
And hopefully that makes intuitive sense as well. And here we just have to think about, well, if I have pi radians, how many degrees is that? Well, that's 180 degrees. It comes straight out of this right over here. Pi radians for every 180 degrees or pi over 180 radians per degree. And this is going to get us to 30 tim... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
It comes straight out of this right over here. Pi radians for every 180 degrees or pi over 180 radians per degree. And this is going to get us to 30 times pi over 180, which we'll simplify to 30 over 180 is 1 over 6. So this is equal to pi over 6. Actually, let me write the units out. This is 30 radians, which is equal... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
So this is equal to pi over 6. Actually, let me write the units out. This is 30 radians, which is equal to pi over 6 radians. Now let's go the other way. Let's think about if we have pi over 3 radians, and I want to convert that to degrees. So what am I going to get if I convert that to degrees? Well, here we're going ... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
Now let's go the other way. Let's think about if we have pi over 3 radians, and I want to convert that to degrees. So what am I going to get if I convert that to degrees? Well, here we're going to want to figure out how many degrees are there per radian. And one way to think about it is, well, think about the pi and th... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
Well, here we're going to want to figure out how many degrees are there per radian. And one way to think about it is, well, think about the pi and the 180. For every 180 degrees, you have pi radians. 180 degrees over pi radians, these are essentially the equivalent thing. Essentially, you're just multiplying this quant... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
180 degrees over pi radians, these are essentially the equivalent thing. Essentially, you're just multiplying this quantity by 1, but you're changing the units. The radians cancel out, and then the pi's cancel out, and you're left with 180 over 3 degrees. 180 over 3 is 60, and we could either write out the word degrees... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
180 over 3 is 60, and we could either write out the word degrees, or you can write degrees just like that. Now let's think about 45 degrees. So what about 45 degrees? And I'll write it like that just so you can figure it out as they're. Figure it out with that notation as well. How many radians will this be equal to? W... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
And I'll write it like that just so you can figure it out as they're. Figure it out with that notation as well. How many radians will this be equal to? Well, once again, we're going to want to think about how many radians do we have per degree. So we're going to multiply this times, well, we know we have pi radians for... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
Well, once again, we're going to want to think about how many radians do we have per degree. So we're going to multiply this times, well, we know we have pi radians for every 180 degrees, or we could even write it this way, pi radians for every 180 degrees. And here, this might be a little less intuitive, the degrees c... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
Actually, let me write this with the words written out. Maybe that's more intuitive when I'm thinking about it in terms of using the notation. So 45 degrees times, we have pi radians for every 180 degrees. So we are left with, when you multiply, 45 times pi over 180, the degrees have canceled out, and you're just left ... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
So we are left with, when you multiply, 45 times pi over 180, the degrees have canceled out, and you're just left with radians, which is equal to what? 45 is half of 90, which is half of 180, so this is 1 4th. This is equal to pi over 4 radians. Let's do one more over here. So let's say that we had negative pi over 2 r... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
Let's do one more over here. So let's say that we had negative pi over 2 radians. What's that going to be in degrees? Well, once again, we have to figure out how many degrees are each of these radians. We know that there are 180 degrees for every pi radians, so we're going to get the radians cancel out, the pi's cancel... | Radian and degree conversion practice Trigonometry Khan Academy.mp3 |
Now what I wanna focus on this video is some features of this graph, and the features we're going to focus on, actually the first of them, is going to be the midline. So pause this video and see if you can figure out the midline of this graph, or the midline of this function, and then we're gonna think about what it ac... | Interpreting trigonometric graphs in context Trigonometry Algebra Khan Academy (2).mp3 |
Well, the extremes are, she goes as low as five and as high as 25. So what's the average of five and 25? Well, that would be 15. So the midline would look something like this, and I'm actually gonna keep going off the graph, and the reason is is to help us think about what does that midline even represent? And one way ... | Interpreting trigonometric graphs in context Trigonometry Algebra Khan Academy (2).mp3 |
So the midline would look something like this, and I'm actually gonna keep going off the graph, and the reason is is to help us think about what does that midline even represent? And one way to think about it is it represents the center of our rotation in this situation, or how high above the ground is the center of ou... | Interpreting trigonometric graphs in context Trigonometry Algebra Khan Academy (2).mp3 |
So I'm going to draw a circle with this as the center, and so the Ferris wheel would look something like, would look something like this, and it has some type of maybe support structure. So the Ferris wheel might look something like that, and this height above the ground, that is 15 meters, that is what the midline is ... | Interpreting trigonometric graphs in context Trigonometry Algebra Khan Academy (2).mp3 |
Pause this video and think about what is the amplitude of this oscillating function right over here, and then we'll think about what does that represent in the real world, or where does it come from in the real world? Well, the amplitude is the maximum difference or the maximum magnitude away from that midline, and you... | Interpreting trigonometric graphs in context Trigonometry Algebra Khan Academy (2).mp3 |
Maybe the Ferris wheel is going this way, at least in my imagination, it's going clockwise, and then after another 10 seconds, she is at 25 meters, so she is right over there, and you can see that. She is right over there. I drew that circle intentionally of that size, and so we see the amplitude in full effect, 10 met... | Interpreting trigonometric graphs in context Trigonometry Algebra Khan Academy (2).mp3 |
Now, the last feature I want to explore is the notion of a period. What is the period of this periodic function? Pause this video and think about that. Well, the period is how much time does it take to complete one cycle? So here, she's starting at the bottom, and let's see, after 10 seconds, not at the bottom yet, aft... | Interpreting trigonometric graphs in context Trigonometry Algebra Khan Academy (2).mp3 |
Well, the period is how much time does it take to complete one cycle? So here, she's starting at the bottom, and let's see, after 10 seconds, not at the bottom yet, after 20 seconds, not at the bottom yet, after 30 seconds, not at the bottom yet, and then here she is, after 40 seconds, she's back at the bottom and abou... | Interpreting trigonometric graphs in context Trigonometry Algebra Khan Academy (2).mp3 |
So pause this video and think about how you would do that. And just to explain how this widget works, if you're trying to do it on Khan Academy, this dot right over here helps define the midline. You can move that up and down. And then this one right over here is a neighboring extreme point. So either a minimum or a ma... | Example Graphing y=-cos(π⋅x)+1.5 Trigonometry Algebra 2 Khan Academy.mp3 |
And then this one right over here is a neighboring extreme point. So either a minimum or a maximum point. So there's a couple of ways that we could approach this. First of all, let's just think about what would cosine of pi x look like? And then we'll think about what the negative does and the plus 1.5. So cosine of pi... | Example Graphing y=-cos(π⋅x)+1.5 Trigonometry Algebra 2 Khan Academy.mp3 |
First of all, let's just think about what would cosine of pi x look like? And then we'll think about what the negative does and the plus 1.5. So cosine of pi x, when x is equal to zero, pi times zero is just going to be zero, cosine of zero is equal to one. And if we're just talking about cosine of pi x, that's going t... | Example Graphing y=-cos(π⋅x)+1.5 Trigonometry Algebra 2 Khan Academy.mp3 |
And if we're just talking about cosine of pi x, that's going to be a maximum point when you hit one. Just cosine of pi x would oscillate between one and negative one. And then what would its period be if we're talking about cosine of pi x? Well, you might remember one way to think about the period is to take two pi and... | Example Graphing y=-cos(π⋅x)+1.5 Trigonometry Algebra 2 Khan Academy.mp3 |
Well, you might remember one way to think about the period is to take two pi and divide it by whatever the coefficient is on the x right over here. So two pi divided by pi would tell us that we have a period of two. And so how do we construct a period of two here? Well, that means that as we start here at x equals zero... | Example Graphing y=-cos(π⋅x)+1.5 Trigonometry Algebra 2 Khan Academy.mp3 |
Well, that means that as we start here at x equals zero, we're at one, we want to get back to that maximum point by the time x is equal to two. So let me see how I can do that. If I were to squeeze it a little bit, that looks pretty good. And the reason why I worked on this midline point is I liked having this maximum ... | Example Graphing y=-cos(π⋅x)+1.5 Trigonometry Algebra 2 Khan Academy.mp3 |
And the reason why I worked on this midline point is I liked having this maximum point at one when x is equal to zero, because we said cosine of pi times zero should be equal to one. So that's why I'm just manipulating this other point in order to set the period right. But this looks right. We're going from this maximu... | Example Graphing y=-cos(π⋅x)+1.5 Trigonometry Algebra 2 Khan Academy.mp3 |
We're going from this maximum point, we're going all the way down and then back to that maximum point, and it looks like our period is indeed two. So this is what the graph of cosine of pi x would look like. Now, what about this negative sign? Well, the negative would essentially flip it around. So instead of whenever ... | Example Graphing y=-cos(π⋅x)+1.5 Trigonometry Algebra 2 Khan Academy.mp3 |
Well, the negative would essentially flip it around. So instead of whenever we're equaling one, we should be equal to negative one. And every time we're equal to negative one, we should be equal to one. So what I could do is I could just take that and then bring it down here. And there you have it, I flipped it around.... | Example Graphing y=-cos(π⋅x)+1.5 Trigonometry Algebra 2 Khan Academy.mp3 |
So what I could do is I could just take that and then bring it down here. And there you have it, I flipped it around. So this is the graph of y equals negative cosine of pi x. And then last but not least, we have this plus 1.5. So that's just going to shift everything up by 1.5. So I'm just gonna shift everything up by... | Example Graphing y=-cos(π⋅x)+1.5 Trigonometry Algebra 2 Khan Academy.mp3 |
And then last but not least, we have this plus 1.5. So that's just going to shift everything up by 1.5. So I'm just gonna shift everything up by, shift it up by 1.5, and shift it up by 1.5. And there you have it. That is the graph of negative cosine of pi x plus 1.5. And you can validate that that's our midline. We're ... | Example Graphing y=-cos(π⋅x)+1.5 Trigonometry Algebra 2 Khan Academy.mp3 |
And there you have it. That is the graph of negative cosine of pi x plus 1.5. And you can validate that that's our midline. We're still oscillating one above and one below. The negative sign, when cosine of pi times zero, that should be one, but then you take the negative of that, we get to negative one. You add 1.5 to... | Example Graphing y=-cos(π⋅x)+1.5 Trigonometry Algebra 2 Khan Academy.mp3 |
So I'm assuming you've had a go at it, and in doing that, you might have realized that, okay, this line, that's one of the sides of this right triangle that I have right over here, and we're given this alpha and beta, but if we consider the combined angle, if we consider the combined angle alpha plus beta, then this si... | Applying angle addition formula for sin Trig identities and examples Trigonometry Khan Academy.mp3 |
So that seems interesting, so let me write that down. Sine of alpha plus beta, sine of alpha plus beta, plus beta, is essentially what we're looking for. Sine of alpha plus beta is this length right over here. Sine of alpha plus beta, it's equal to the opposite side, that, over the hypotenuse. Well, the hypotenuse is j... | Applying angle addition formula for sin Trig identities and examples Trigonometry Khan Academy.mp3 |
Sine of alpha plus beta, it's equal to the opposite side, that, over the hypotenuse. Well, the hypotenuse is just going to be equal to one, so it's equal to this side. So another way of phrasing the exact same problem that we first tried to tackle is how do we figure out the sine of alpha plus beta? And if you're famil... | Applying angle addition formula for sin Trig identities and examples Trigonometry Khan Academy.mp3 |
And if you're familiar with your trig identities, something might be jumping out at you, that hey, we know a different way of expressing sine of alpha plus beta. We know that this thing is the same thing as, we know it's the same thing as the sine of alpha, plus, or sine of alpha times the cosine of beta, plus the othe... | Applying angle addition formula for sin Trig identities and examples Trigonometry Khan Academy.mp3 |
Let me draw a line here so we don't get confused. So if we're trying to figure this out, and we know that this can be re-expressed this way, it all boils down to can we figure out what sine of alpha is, cosine of beta, cosine of alpha, and sine of beta? And when you look at this, you see that you actually can figure th... | Applying angle addition formula for sin Trig identities and examples Trigonometry Khan Academy.mp3 |
So let's do that. Sine of alpha, I'll write it over here, sine of alpha is equal to, this is alpha, sine is opposite over hypotenuse, so it's 0.5 over one. So this is equal to 0.5. So that is 0.5. Cosine of beta, cosine of beta, this is beta, cosine is adjacent over hypotenuse. So this is beta, the adjacent side is 0.6... | Applying angle addition formula for sin Trig identities and examples Trigonometry Khan Academy.mp3 |
So that is 0.5. Cosine of beta, cosine of beta, this is beta, cosine is adjacent over hypotenuse. So this is beta, the adjacent side is 0.6 over the hypotenuse of one, so it's 0.6. 0.6, 0.6. Cosine of alpha, cosine of alpha, adjacent over hypotenuse, it's square root of three over two over one. So that's just square ro... | Applying angle addition formula for sin Trig identities and examples Trigonometry Khan Academy.mp3 |
0.6, 0.6. Cosine of alpha, cosine of alpha, adjacent over hypotenuse, it's square root of three over two over one. So that's just square root of three over two. So this is just square root of three over two, and then finally, sine of beta, sine of beta, opposite over hypotenuse, is 0.8. This is 0.8. And actually, let m... | Applying angle addition formula for sin Trig identities and examples Trigonometry Khan Academy.mp3 |
So this is just square root of three over two, and then finally, sine of beta, sine of beta, opposite over hypotenuse, is 0.8. This is 0.8. And actually, let me write that as, I'm gonna write that as 4 5ths, just so that that's the same thing as 0.8, just because I think it's gonna make it a little bit easier for me to... | Applying angle addition formula for sin Trig identities and examples Trigonometry Khan Academy.mp3 |
So what is all of this equal to? Well, this is going to be equal to 0.5 times 0.6, this part right over here is 0.3, 0.3, and square root of three over two times 4 5ths, well, let's just multiply them. Well, four divided by two is two, so it's two square roots of three over five. So this is equal to, or so plus, two sq... | Applying angle addition formula for sin Trig identities and examples Trigonometry Khan Academy.mp3 |
So I've got my little scratch pad here to try to work that through. So let's figure out what g inverse of x is. This is g of x. So g inverse of x, I'm essentially, let me just read, this is g of x right over here. g of x is equal to tangent of x minus 3 pi over 2 plus 6. So g inverse of x, I essentially can swap the, I... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So g inverse of x, I'm essentially, let me just read, this is g of x right over here. g of x is equal to tangent of x minus 3 pi over 2 plus 6. So g inverse of x, I essentially can swap the, I can replace the x with the g inverse of x and replace the g of x with an x, and then solve for g inverse of x. So I could write... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So I could write that x is equal to tangent of g inverse of x minus 3 pi over 2 plus 6. So let's just solve for g inverse of x. And I actually encourage you to pause this video and try to work through this out, or work it out on your own. So let's subtract 6 from both sides to at least get rid of this 6 here. And so I'... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So let's subtract 6 from both sides to at least get rid of this 6 here. And so I'm left with x minus 6 is equal to the tangent of g inverse of x minus 3 pi over 2. Now let's take the inverse tangent of both sides of this equation. So the inverse tangent of the left-hand side is the inverse tangent of x minus 6. And on ... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So the inverse tangent of the left-hand side is the inverse tangent of x minus 6. And on the right-hand side, the inverse tangent of tangent, if we restrict the domain in the proper way, and we'll talk about that in a little bit, is just going to be what the input into the tangent function is. So if you restrict the do... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
Because again, if we restrict the domain, if we restrict what the possible values of theta are in the right way. So let's just assume that we're doing that. And so the inverse tangent of the tan of this is going to be just this stuff right over here. It's just going to be that. It's going to be g inverse of x minus 3 p... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
It's just going to be that. It's going to be g inverse of x minus 3 pi over 2. And now we're in the home stretch. To solve for g inverse of x, we could just add 3 pi over 2 to both sides. So we get, and actually let me just swap both sides. We get g inverse of x is equal to the inverse tangent of x minus 6. And then we... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
To solve for g inverse of x, we could just add 3 pi over 2 to both sides. So we get, and actually let me just swap both sides. We get g inverse of x is equal to the inverse tangent of x minus 6. And then we're adding 3 pi over 2 to both sides. So this side is now on this side, so plus 3 pi over 2. So let me actually ty... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
And then we're adding 3 pi over 2 to both sides. So this side is now on this side, so plus 3 pi over 2. So let me actually type that in. Let me see if I can remember it, because I'm about to lose this on my screen. So inverse tangent of x minus 6 plus 3 pi over 2. So let me write that down. So let me type this. | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
Let me see if I can remember it, because I'm about to lose this on my screen. So inverse tangent of x minus 6 plus 3 pi over 2. So let me write that down. So let me type this. So g inverse of x is going to be the inverse tangent. So I could write it like this. The inverse tangent of x minus 6. | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So let me type this. So g inverse of x is going to be the inverse tangent. So I could write it like this. The inverse tangent of x minus 6. And yes, it interpreted it correctly. Inverse tangent, you can view that as arc tangent of x minus 6 plus 3 pi over 2. And it did interpret it correctly. | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
The inverse tangent of x minus 6. And yes, it interpreted it correctly. Inverse tangent, you can view that as arc tangent of x minus 6 plus 3 pi over 2. And it did interpret it correctly. But then we have to think about what is the domain of g inverse? What is the domain of g inverse of x? Let's think about this a litt... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
And it did interpret it correctly. But then we have to think about what is the domain of g inverse? What is the domain of g inverse of x? Let's think about this a little bit more. The domain of g inverse of x. So let's just think about what tangent is doing. So the tangent function, if we imagine a unit circle, so that... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
Let's think about this a little bit more. The domain of g inverse of x. So let's just think about what tangent is doing. So the tangent function, if we imagine a unit circle, so that's a unit circle right over there. I guess we can imagine it to be a unit circle. My pen tool is acting up a little bit. I'm putting these... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So the tangent function, if we imagine a unit circle, so that's a unit circle right over there. I guess we can imagine it to be a unit circle. My pen tool is acting up a little bit. I'm putting these little gaps and things, but I think we can power through that. So let's just say, for the sake of argument, that that's ... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
I'm putting these little gaps and things, but I think we can power through that. So let's just say, for the sake of argument, that that's a unit circle. That's the x-axis and that's the y-axis. If you form an angle theta, if you form some angle theta right over here, the tangent of theta is essentially the slope of thi... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
If you form an angle theta, if you form some angle theta right over here, the tangent of theta is essentially the slope of this terminal ray of the angle. Or I guess you could call it the terminal ray of the angle. The angle is formed by that ray and this ray along the positive x-axis. So the tangent of theta is the sl... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So the tangent of theta is the slope right over there. And you can get a tangent of any theta except for a few. So you can find the tangent of that. You could find the slope there. You could find the slope there. You could also find the slope there. You could find the slope there. | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
You could find the slope there. You could find the slope there. You could also find the slope there. You could find the slope there. But the place where you can't find the slope is when this ray goes straight up or this ray goes straight down. Those are the cases where you can't find the slope. There the slope you coul... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
You could find the slope there. But the place where you can't find the slope is when this ray goes straight up or this ray goes straight down. Those are the cases where you can't find the slope. There the slope you could say is approaching positive or negative infinity. So the domain of tangent is essentially all real ... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
There the slope you could say is approaching positive or negative infinity. So the domain of tangent is essentially all real numbers, all reals except pi over 2 plus multiples of pi. Where k could be any integer. So you could also be subtracting pi because if you have pi over 2, if you add pi, you go straight down here... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So you could also be subtracting pi because if you have pi over 2, if you add pi, you go straight down here. You add another pi, you go up there. If you subtract pi, you go down here. Subtract another pi, you go over there. So this is the domain. But given this domain, you can get any real number. So the range here is ... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
Subtract another pi, you go over there. So this is the domain. But given this domain, you can get any real number. So the range here is all reals because you can get any slope here. You can increase theta if you want a really high slope, decrease theta if you want a really negative slope right over there. So you can re... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So the range here is all reals because you can get any slope here. You can increase theta if you want a really high slope, decrease theta if you want a really negative slope right over there. So you can really get to anything. Now when you're talking about the inverse tangent, by convention you're going to make tangent... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
Now when you're talking about the inverse tangent, by convention you're going to make tangent invertible so that you don't have multiple elements of your domain all mapping to the same element of the range. Because for example, this angle right over here has the exact same slope as this angle right over here. So if you... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So the convention is that to make tangent invertible, you restrict its domain. So you restrict the domain to the interval from negative pi over 2 to pi over 2 in order to construct the inverse tangent. So the inverse tangent, you can input any real number into it. So the inverse tangent's domain, this is just a convent... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So the inverse tangent's domain, this is just a convention, they could have restricted tangent's domains as long as for any theta, there's only one theta in that domain that maps to a specific element of the range. But the convention is to restrict tangent's domain between negative pi over 2 and pi over 2. So inverse t... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
Its range, and this is by convention, it's going to be between negative pi over 2 and pi over 2 and not including them. So let's go back to our original question right over here. What is the domain of G inverse? So let's look at the domain of G inverse. Well, G inverse, the domain of this, I could put any real number i... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So let's look at the domain of G inverse. Well, G inverse, the domain of this, I could put any real number in here. Any real number here. Now, what this is going to pop out is going to be something between negative pi over 2 and pi over 2, but they're not asking us the range of G inverse. Actually, it would have been a... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
Now, what this is going to pop out is going to be something between negative pi over 2 and pi over 2, but they're not asking us the range of G inverse. Actually, it would have been a more interesting question. They're asking us what's the domain of G inverse, and I could put in any real number right here for x. So let'... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So let's put that in here. So the domain of G inverse of x, it's negative infinity to infinity. But actually, just for fun, and let's just verify that we got the question right, and we did. But just for fun, actually, I am curious. Let's just think about what the range of G inverse is. So the range of this thing right ... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
But just for fun, actually, I am curious. Let's just think about what the range of G inverse is. So the range of this thing right over here is going to be between negative pi over 2 to pi over 2. That's for this part right over here. And then we're going to add 3 pi over 2s to it. So the range for the entire function, ... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
That's for this part right over here. And then we're going to add 3 pi over 2s to it. So the range for the entire function, so the range for this thing, the range is going to be what the low end, if we add 3 pi over 2 to this, this is going to give us 2 pi over 2, which is just going to be, so 3 pi over 2 minus pi over... | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
So the range of G inverse of x is pi to 2 pi, and it's an open interval, it doesn't include the boundaries, but its domain, you could put any value for x here, and it will be defined. | Inverse tan domain and range Trigonometry Khan Academy.mp3 |
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