video_title
stringlengths 55
104
| Sentence
stringlengths 141
2.08k
|
|---|---|
Introduction to the unit circle Trigonometry Khan Academy.mp3 | The length of the adjacent side, for this angle, the adjacent side has length a, so it's going to be equal to a, over what's the length of the hypotenuse? Well, that's just one. So the cosine of theta is just equal to a. Let me write this down again. So the cosine of theta is just equal to a. It's equal to the x-coordinate of where this terminal side of the angle intersected the unit circle. Now let's think about the sine of theta. |
Introduction to the unit circle Trigonometry Khan Academy.mp3 | Let me write this down again. So the cosine of theta is just equal to a. It's equal to the x-coordinate of where this terminal side of the angle intersected the unit circle. Now let's think about the sine of theta. Sine of theta, and I'm going to do it in, well, let me see, I'll do it in orange. So what's the sine of theta going to be? Well, we just have to look at the Soh part of our Soh-Cah-Toa definition. |
Introduction to the unit circle Trigonometry Khan Academy.mp3 | Now let's think about the sine of theta. Sine of theta, and I'm going to do it in, well, let me see, I'll do it in orange. So what's the sine of theta going to be? Well, we just have to look at the Soh part of our Soh-Cah-Toa definition. It tells us that sine is opposite over hypotenuse. Well, the opposite side here has length b, and the hypotenuse has length one. So our sine of theta is equal to b. |
Introduction to the unit circle Trigonometry Khan Academy.mp3 | Well, we just have to look at the Soh part of our Soh-Cah-Toa definition. It tells us that sine is opposite over hypotenuse. Well, the opposite side here has length b, and the hypotenuse has length one. So our sine of theta is equal to b. So an interesting thing, this coordinate, this point where our terminal side of our angle intersected the unit circle, that point a, b, we could also view this as a is the same thing as cosine of theta. A is the same thing as cosine of theta, and b is the same thing as sine of theta. Well, that's interesting. |
Introduction to the unit circle Trigonometry Khan Academy.mp3 | So our sine of theta is equal to b. So an interesting thing, this coordinate, this point where our terminal side of our angle intersected the unit circle, that point a, b, we could also view this as a is the same thing as cosine of theta. A is the same thing as cosine of theta, and b is the same thing as sine of theta. Well, that's interesting. That was just, we just used our Soh-Cah-Toa definition. Now, can we in some way use this to extend Soh-Cah-Toa? Because Soh-Cah-Toa has a problem. |
Introduction to the unit circle Trigonometry Khan Academy.mp3 | Well, that's interesting. That was just, we just used our Soh-Cah-Toa definition. Now, can we in some way use this to extend Soh-Cah-Toa? Because Soh-Cah-Toa has a problem. It works out fine if our angle is greater than zero degrees, if we're dealing with degrees, and if it's less than 90 degrees. We can always make it part of a right triangle. But Soh-Cah-Toa starts to break down as our angle is either zero, or maybe even becomes negative, or as our angle is 90 degrees or more. |
Introduction to the unit circle Trigonometry Khan Academy.mp3 | Because Soh-Cah-Toa has a problem. It works out fine if our angle is greater than zero degrees, if we're dealing with degrees, and if it's less than 90 degrees. We can always make it part of a right triangle. But Soh-Cah-Toa starts to break down as our angle is either zero, or maybe even becomes negative, or as our angle is 90 degrees or more. You can't have a right triangle with two 90 degree angles in it. It starts to break down. Let me make this clear. |
Introduction to the unit circle Trigonometry Khan Academy.mp3 | But Soh-Cah-Toa starts to break down as our angle is either zero, or maybe even becomes negative, or as our angle is 90 degrees or more. You can't have a right triangle with two 90 degree angles in it. It starts to break down. Let me make this clear. So sure, that's, so this is a right triangle, so the angle is pretty large. I can make the angle even larger, and still have a right triangle even larger. But I can never get quite to 90 degrees. |
Introduction to the unit circle Trigonometry Khan Academy.mp3 | Let me make this clear. So sure, that's, so this is a right triangle, so the angle is pretty large. I can make the angle even larger, and still have a right triangle even larger. But I can never get quite to 90 degrees. At 90 degrees, it's not clear that I have a right triangle anymore. It all seems to break down, and especially the case what happens when I go beyond 90 degrees. So let's go, let's see if we can use what we set up here. |
Introduction to the unit circle Trigonometry Khan Academy.mp3 | But I can never get quite to 90 degrees. At 90 degrees, it's not clear that I have a right triangle anymore. It all seems to break down, and especially the case what happens when I go beyond 90 degrees. So let's go, let's see if we can use what we set up here. Let's set up a new definition of our trig functions, which is really an extension of Soh-Cah-Toa, and it's consistent with Soh-Cah-Toa. Instead of defining cosine as, oh, if I have a right triangle, and saying, okay, it's the adjacent over the hypotenuse, sine is the opposite over the hypotenuse, tangent is opposite over adjacent, why don't I just say, for any angle, I can draw it in the unit circle using this convention that I just set up, and let's just say that the cosine of our angle, the cosine of our angle is equal to the x-coordinate where we intersect, is equal to the x-coordinate, coordinate where the terminal side of our angle intersects the unit circle, where angle, I'll write the terminal side, terminal side of angle, side of angle intersects, intersects the unit, the unit circle, and why don't we define sine of theta, sine of theta to be equal to the y-coordinate, where the terminal side of the angle intersects the unit circle. So essentially, for any angle, this point is going to define cosine of theta and sine of theta, and so what would be the reasonable definition for tangent of theta? |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | We're asked to graph the function y is equal to 2 sine of negative x on the interval, the closed interval, so it includes the endpoints, negative 2 pi to 2 pi. So to do this, I'm first going to graph the function y is equal to sine of x and then think about how it's changed by the 2 and this negative in front of the x right over here. So let's do y equal to sine of x first. So let me draw our x-axis. Let me draw the y-axis. Pretty straightforward. And we care about it between negative 2 pi and 2 pi. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So let me draw our x-axis. Let me draw the y-axis. Pretty straightforward. And we care about it between negative 2 pi and 2 pi. So let's say that this is negative 2 pi, and then this right over here would be negative pi. This, of course, is 0. Then this is positive pi, and then this right over here is 2 pi again. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | And we care about it between negative 2 pi and 2 pi. So let's say that this is negative 2 pi, and then this right over here would be negative pi. This, of course, is 0. Then this is positive pi, and then this right over here is 2 pi again. This right over here is 2 pi, and then this could be 1, this could be 2, this could be negative 1, and this could be negative 2. And let me copy this thing so I can use it for later when I adjust this graph. So let me copy. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | Then this is positive pi, and then this right over here is 2 pi again. This right over here is 2 pi, and then this could be 1, this could be 2, this could be negative 1, and this could be negative 2. And let me copy this thing so I can use it for later when I adjust this graph. So let me copy. All right. So let's think about sine of x. So what happens when sine is 0? |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So let me copy. All right. So let's think about sine of x. So what happens when sine is 0? When sine is 0, or sorry, when x is 0, sine of 0 is 0. And I'll draw a little unit circle here for reference. This is what I like to do in my head as I'm trying to figure out the value of the trigonometric functions. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So what happens when sine is 0? When sine is 0, or sorry, when x is 0, sine of 0 is 0. And I'll draw a little unit circle here for reference. This is what I like to do in my head as I'm trying to figure out the value of the trigonometric functions. So this is x, this is y. Draw a unit circle. And remember, over here, x is referring to the angle. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | This is what I like to do in my head as I'm trying to figure out the value of the trigonometric functions. So this is x, this is y. Draw a unit circle. And remember, over here, x is referring to the angle. So that's my unit circle, radius 1. So when the angle is 0, sine is going to be the y-coordinate here. So sine of 0 is 0. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | And remember, over here, x is referring to the angle. So that's my unit circle, radius 1. So when the angle is 0, sine is going to be the y-coordinate here. So sine of 0 is 0. When, as sine increases, we go up all the way to sine of pi over 2 is 1. So sine of pi over 2 is going to get you to 1. Then you go sine of pi gets you to 0. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So sine of 0 is 0. When, as sine increases, we go up all the way to sine of pi over 2 is 1. So sine of pi over 2 is going to get you to 1. Then you go sine of pi gets you to 0. Sine of 3 pi over 2 gets you to negative 1. And then sine of 2 pi gets you back to 0. So if I were to graph it, it looks something like this. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | Then you go sine of pi gets you to 0. Sine of 3 pi over 2 gets you to negative 1. And then sine of 2 pi gets you back to 0. So if I were to graph it, it looks something like this. So this is between 0 and 2 pi. It looks something like that. And we also want to go in the negative direction. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So if I were to graph it, it looks something like this. So this is between 0 and 2 pi. It looks something like that. And we also want to go in the negative direction. So as we go in the negative direction, so sine of negative pi over 2 is negative 1. Negative pi over 2 is negative 1. Then you go back to negative pi, go back to 0. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | And we also want to go in the negative direction. So as we go in the negative direction, so sine of negative pi over 2 is negative 1. Negative pi over 2 is negative 1. Then you go back to negative pi, go back to 0. Negative 3 pi over 2, you're going all the way around like that. That gets you back to sine is equal to 1. So sine is equal to 1. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | Then you go back to negative pi, go back to 0. Negative 3 pi over 2, you're going all the way around like that. That gets you back to sine is equal to 1. So sine is equal to 1. And then you go 2 pi, sine is back, is equaling 0. So the curve will look something like this in the negative. So as we go from between 0 and negative 2 pi. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So sine is equal to 1. And then you go 2 pi, sine is back, is equaling 0. So the curve will look something like this in the negative. So as we go from between 0 and negative 2 pi. This is consistent with everything else that we know about sine. The period of sine of x, well you see here you have a coefficient of 1 here. So the period is just going to be 2 pi over the absolute value of 1, which is a little bit obvious. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So as we go from between 0 and negative 2 pi. This is consistent with everything else that we know about sine. The period of sine of x, well you see here you have a coefficient of 1 here. So the period is just going to be 2 pi over the absolute value of 1, which is a little bit obvious. It's just 2 pi. Or you just see here that the period was 2 pi. It took 2 pi length to do one of our smallest repeating pattern right over here. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So the period is just going to be 2 pi over the absolute value of 1, which is a little bit obvious. It's just 2 pi. Or you just see here that the period was 2 pi. It took 2 pi length to do one of our smallest repeating pattern right over here. And what is the amplitude? Well, we vary between 1 and negative 1. The total difference between the minimum and the maximum is 2. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | It took 2 pi length to do one of our smallest repeating pattern right over here. And what is the amplitude? Well, we vary between 1 and negative 1. The total difference between the minimum and the maximum is 2. Half of that is 1. Or another way of thinking about it, we vary 1 from our middle point. So that was pretty straightforward. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | The total difference between the minimum and the maximum is 2. Half of that is 1. Or another way of thinking about it, we vary 1 from our middle point. So that was pretty straightforward. Let's change it up a little bit. Now let's graph y is equal to 2 sine of x. So let me put my little axes there. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So that was pretty straightforward. Let's change it up a little bit. Now let's graph y is equal to 2 sine of x. So let me put my little axes there. I want to do it right under it. And so what is going to happen now that we have y is equal to 2 sine of x? How is the graph going to change? |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So let me put my little axes there. I want to do it right under it. And so what is going to happen now that we have y is equal to 2 sine of x? How is the graph going to change? Well, all we did is we multiplied this function by 2. So whatever values it takes on, it's going to be twice as high now. So 2 times 0 is 0. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | How is the graph going to change? Well, all we did is we multiplied this function by 2. So whatever values it takes on, it's going to be twice as high now. So 2 times 0 is 0. 2 times 1 is now 2. 2 times 1 is 2. 2 times 0 is 2. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So 2 times 0 is 0. 2 times 1 is now 2. 2 times 1 is 2. 2 times 0 is 2. That's at pi over 2. 2 times 0 is 0. 2 times negative 1 is negative 2. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | 2 times 0 is 2. That's at pi over 2. 2 times 0 is 0. 2 times negative 1 is negative 2. 2 times 0 is 0. So it looks something like this between 0 and 2 pi. It looks something like that. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | 2 times negative 1 is negative 2. 2 times 0 is 0. So it looks something like this between 0 and 2 pi. It looks something like that. And we could keep going in the negative direction. 2 times negative 1 is negative 2. 2 times 0 is 0. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | It looks something like that. And we could keep going in the negative direction. 2 times negative 1 is negative 2. 2 times 0 is 0. 2 times 1 is positive 2. 2 times 0 is 0. So in the negative direction, it looks something like that. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | 2 times 0 is 0. 2 times 1 is positive 2. 2 times 0 is 0. So in the negative direction, it looks something like that. My best attempt to draw a relatively smooth curve. Hopefully you get the idea. So it will look something like that. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So in the negative direction, it looks something like that. My best attempt to draw a relatively smooth curve. Hopefully you get the idea. So it will look something like that. So what just happened? Well, the difference between the minimum value and the maximum value just increased by a factor of 2. The total difference is 4. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So it will look something like that. So what just happened? Well, the difference between the minimum value and the maximum value just increased by a factor of 2. The total difference is 4. Half of that difference is now 2. So what is the amplitude here? Well, the amplitude is 2. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | The total difference is 4. Half of that difference is now 2. So what is the amplitude here? Well, the amplitude is 2. So the amplitude, you could view it as the absolute value of this thing. The absolute value of 2 is now equal to 2. And it's common sense. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | Well, the amplitude is 2. So the amplitude, you could view it as the absolute value of this thing. The absolute value of 2 is now equal to 2. And it's common sense. The amplitude here was 1. And now you're swaying from that middle position twice as far because you're multiplying by 2. Now let's go back to sine of x. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | And it's common sense. The amplitude here was 1. And now you're swaying from that middle position twice as far because you're multiplying by 2. Now let's go back to sine of x. And let's change it in a different way. Let's graph sine of negative x. So let me once again put some graph paper here. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | Now let's go back to sine of x. And let's change it in a different way. Let's graph sine of negative x. So let me once again put some graph paper here. And now my goal is to graph y is equal to sine of negative x. So for at least the time being, I got rid of that 2 there. And I'm just going straight from sine of x to sine of negative x. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So let me once again put some graph paper here. And now my goal is to graph y is equal to sine of negative x. So for at least the time being, I got rid of that 2 there. And I'm just going straight from sine of x to sine of negative x. So let's think about how the values are going to work out. So when x is 0, this is still going to be sine of 0, which is equal to 0. But then what happens as x increases? |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | And I'm just going straight from sine of x to sine of negative x. So let's think about how the values are going to work out. So when x is 0, this is still going to be sine of 0, which is equal to 0. But then what happens as x increases? What happens when x is pi over 2? When x is pi over 2, the angle that we're inputting into sine, we're going to have to multiply it by this negative. So when x is pi over 2, we're really taking sine of negative pi over 2. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | But then what happens as x increases? What happens when x is pi over 2? When x is pi over 2, the angle that we're inputting into sine, we're going to have to multiply it by this negative. So when x is pi over 2, we're really taking sine of negative pi over 2. Well, what's sine of negative pi over 2? Well, we could see that right over here. It's negative 1. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So when x is pi over 2, we're really taking sine of negative pi over 2. Well, what's sine of negative pi over 2? Well, we could see that right over here. It's negative 1. And then when x is equal to pi, well, sine of negative pi we already see is 0. When x is 3 pi over 2, well, it's going to be sine of negative 3 pi over 2, which is 1. And once again, when x is 2 pi, it's going to be sine of negative 2 pi. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | It's negative 1. And then when x is equal to pi, well, sine of negative pi we already see is 0. When x is 3 pi over 2, well, it's going to be sine of negative 3 pi over 2, which is 1. And once again, when x is 2 pi, it's going to be sine of negative 2 pi. Sine of negative 2 pi is 0. So notice what was happening as I was trying to graph between 0 and 2 pi. I kept referring to the points in the negative direction. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | And once again, when x is 2 pi, it's going to be sine of negative 2 pi. Sine of negative 2 pi is 0. So notice what was happening as I was trying to graph between 0 and 2 pi. I kept referring to the points in the negative direction. So you can imagine taking this negative side right over here, between 0 and negative 2 pi, and then flipping it over to get this one right over here. That's what that negative x seemed to do. And by that same logic, when we go in the negative direction, you say when x is equal to negative pi over 2, well, you have that negative in front of it, so it's going to be sine of pi over 2. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | I kept referring to the points in the negative direction. So you can imagine taking this negative side right over here, between 0 and negative 2 pi, and then flipping it over to get this one right over here. That's what that negative x seemed to do. And by that same logic, when we go in the negative direction, you say when x is equal to negative pi over 2, well, you have that negative in front of it, so it's going to be sine of pi over 2. Well, it's going to be equal to 1. And then you can flip this over the y-axis. So essentially what we have done is we have flipped it. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | And by that same logic, when we go in the negative direction, you say when x is equal to negative pi over 2, well, you have that negative in front of it, so it's going to be sine of pi over 2. Well, it's going to be equal to 1. And then you can flip this over the y-axis. So essentially what we have done is we have flipped it. We have reflected the graph of sine of x over the y-axis. This is the y-axis here, of course. So we have reflected it over the y-axis. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So essentially what we have done is we have flipped it. We have reflected the graph of sine of x over the y-axis. This is the y-axis here, of course. So we have reflected it over the y-axis. So let me make sure I'm so it looks something like this. This is the y-axis. Hopefully you see that reflection right over there. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So we have reflected it over the y-axis. So let me make sure I'm so it looks something like this. This is the y-axis. Hopefully you see that reflection right over there. That's what that negative x has done. So now let's think about kind of the combo, having the 2 out the front and the negative x right over there. So let me put our graph, my little axes there one more time. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | Hopefully you see that reflection right over there. That's what that negative x has done. So now let's think about kind of the combo, having the 2 out the front and the negative x right over there. So let me put our graph, my little axes there one more time. And now let's try to do what was asked of us. So I'll do it in a new color. I'll do it in blue. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So let me put our graph, my little axes there one more time. And now let's try to do what was asked of us. So I'll do it in a new color. I'll do it in blue. Now let's graph. This is our y-axis. This is a y-axis. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | I'll do it in blue. Now let's graph. This is our y-axis. This is a y-axis. Let's graph y is equal to 2 times sine of negative x. So based on everything we've done, how will this look? What are the transformations we would do if we're going from original sine of x to y is equal to 2 sine of negative x? |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | This is a y-axis. Let's graph y is equal to 2 times sine of negative x. So based on everything we've done, how will this look? What are the transformations we would do if we're going from original sine of x to y is equal to 2 sine of negative x? Well, there's two ways you could think about it. You could either take 2 sine of x. So here we multiplied by 2 to get double the amplitude. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | What are the transformations we would do if we're going from original sine of x to y is equal to 2 sine of negative x? Well, there's two ways you could think about it. You could either take 2 sine of x. So here we multiplied by 2 to get double the amplitude. And you could say, well, I'm going to now flip it over to get the negative sine of x. And so if you did that, you would get, so let me make it clear what I'm flipping. So if I took between 0 and negative 2 pi and I were to flip it over, what used to be here, you flip it over, you reflect it over the y-axis, and you now have, so it'll go negative first. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So here we multiplied by 2 to get double the amplitude. And you could say, well, I'm going to now flip it over to get the negative sine of x. And so if you did that, you would get, so let me make it clear what I'm flipping. So if I took between 0 and negative 2 pi and I were to flip it over, what used to be here, you flip it over, you reflect it over the y-axis, and you now have, so it'll go negative first. Then it'll go back to 0. Then it'll go positive. And then you get right over there. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So if I took between 0 and negative 2 pi and I were to flip it over, what used to be here, you flip it over, you reflect it over the y-axis, and you now have, so it'll go negative first. Then it'll go back to 0. Then it'll go positive. And then you get right over there. So all I did to go from 2 sine of x to 2 sine of negative x is I just reflected over the y-axis. And then, of course, what is between 0 and negative 2 pi? You just have to look between 0 and 2 pi. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | And then you get right over there. So all I did to go from 2 sine of x to 2 sine of negative x is I just reflected over the y-axis. And then, of course, what is between 0 and negative 2 pi? You just have to look between 0 and 2 pi. So now it's going to go up and down. Let me make it a little bit better, draw it a little bit neater. And then down and up. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | You just have to look between 0 and 2 pi. So now it's going to go up and down. Let me make it a little bit better, draw it a little bit neater. And then down and up. So it was a reflection of what was between 0 and 2 pi. So you see that right over here. Or if you start with sine of negative x and you go to 2 sine of negative x, notice what happens between sine of negative x and 2 sine of negative x. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | And then down and up. So it was a reflection of what was between 0 and 2 pi. So you see that right over here. Or if you start with sine of negative x and you go to 2 sine of negative x, notice what happens between sine of negative x and 2 sine of negative x. What's the difference between this graph and this graph? Well, we just have twice the amplitude. We're multiplying this one by 2, and so you get twice the amplitude. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | Or if you start with sine of negative x and you go to 2 sine of negative x, notice what happens between sine of negative x and 2 sine of negative x. What's the difference between this graph and this graph? Well, we just have twice the amplitude. We're multiplying this one by 2, and so you get twice the amplitude. So the last thought or question I have for you is how does the period of 2 sine of negative x relate to the period of sine of x? Well, there's two ways to think about it. Actually, I'll let you think about that for a second. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | We're multiplying this one by 2, and so you get twice the amplitude. So the last thought or question I have for you is how does the period of 2 sine of negative x relate to the period of sine of x? Well, there's two ways to think about it. Actually, I'll let you think about that for a second. Well, there's two ways to think about it. You could refer to the graphs right over here, or you could think about the formula, which might be a little bit intuitive right now. If you wanted to refer to the kind of classical formula, you would say the period is just going to be 2 pi, and you divide by the absolute value of the coefficient to figure out how much faster are you going to get to 2 pi right over here. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | Actually, I'll let you think about that for a second. Well, there's two ways to think about it. You could refer to the graphs right over here, or you could think about the formula, which might be a little bit intuitive right now. If you wanted to refer to the kind of classical formula, you would say the period is just going to be 2 pi, and you divide by the absolute value of the coefficient to figure out how much faster are you going to get to 2 pi right over here. So the absolute value of negative x, or the absolute value of the negative 1, is just 1. So you get 2 pi. So you have the exact same period as the period of sine of x. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | If you wanted to refer to the kind of classical formula, you would say the period is just going to be 2 pi, and you divide by the absolute value of the coefficient to figure out how much faster are you going to get to 2 pi right over here. So the absolute value of negative x, or the absolute value of the negative 1, is just 1. So you get 2 pi. So you have the exact same period as the period of sine of x. And if you see that, you complete one cycle every 2 pi. Now, what is the difference? Well, the period's the same, but remember, this negative x, it's not like you can just completely ignore it. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | So you have the exact same period as the period of sine of x. And if you see that, you complete one cycle every 2 pi. Now, what is the difference? Well, the period's the same, but remember, this negative x, it's not like you can just completely ignore it. It doesn't change the period, but it does change how the graph looks. When you start getting increased x's, instead of sine being positive, as it would be in the case of the traditional sine function, as x grows, you're taking the sine of negative x. You're taking the sine of a negative angle, and so that's why you start off having negative values of sine. |
Example Amplitude and period transformations Trigonometry Khan Academy.mp3 | Well, the period's the same, but remember, this negative x, it's not like you can just completely ignore it. It doesn't change the period, but it does change how the graph looks. When you start getting increased x's, instead of sine being positive, as it would be in the case of the traditional sine function, as x grows, you're taking the sine of negative x. You're taking the sine of a negative angle, and so that's why you start off having negative values of sine. And that's also another way, if you want to think about it, that it's the reflection over the y-axis of just sine of x. These two are reflections, and these two are reflections. This one is 2 times the amplitude as this one. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | So what does that mean? So we have these two terms, and I want to figure out their greatest common monomial factor, and then I want to express this with that greatest common monomial factor factored out. So how can we tackle it? Well, one way to start is I can look at just the constant terms. I can look at, or not the constants, the coefficients, I should say. So I have the eight and the 12, and I can say, well, what is just the greatest common factor of eight and 12? The GCF of eight and 12. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | Well, one way to start is I can look at just the constant terms. I can look at, or not the constants, the coefficients, I should say. So I have the eight and the 12, and I can say, well, what is just the greatest common factor of eight and 12? The GCF of eight and 12. And there are a lot of common factors of eight and 12. They're both divisible by one, they're both divisible by two, they're both divisible by four, but the greatest of their common factors is going to be four. So that is equal to four. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | The GCF of eight and 12. And there are a lot of common factors of eight and 12. They're both divisible by one, they're both divisible by two, they're both divisible by four, but the greatest of their common factors is going to be four. So that is equal to four. So let me just leave that there. And then we can think about what is, well, let me actually write it right over here. I'll put a four here. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | So that is equal to four. So let me just leave that there. And then we can think about what is, well, let me actually write it right over here. I'll put a four here. And now we can move on to the powers of x. We have an x squared and we have an x. And we can say, what is the largest power of x that is divisible into both x squared and x? |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | I'll put a four here. And now we can move on to the powers of x. We have an x squared and we have an x. And we can say, what is the largest power of x that is divisible into both x squared and x? Well, that's just going to be x. X squared is clearly divisible by x, and x is clearly divisible by x, but x isn't going to be, isn't going to have a larger power of x as a factor. So this is the greatest, you could view this as the greatest common monomial factor of x squared and x. Now we do the same thing for the y's. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | And we can say, what is the largest power of x that is divisible into both x squared and x? Well, that's just going to be x. X squared is clearly divisible by x, and x is clearly divisible by x, but x isn't going to be, isn't going to have a larger power of x as a factor. So this is the greatest, you could view this as the greatest common monomial factor of x squared and x. Now we do the same thing for the y's. So we have a y and a y squared. If we think in the same terms, the largest power of y that's divisible into both of these is going to be just y to the first power, or y. And so four xy is the greatest common monomial factor. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | Now we do the same thing for the y's. So we have a y and a y squared. If we think in the same terms, the largest power of y that's divisible into both of these is going to be just y to the first power, or y. And so four xy is the greatest common monomial factor. And to see that, we can express each of these terms as a product of four xy and something else. So this first term right over here, so let me pick a color. So this term right over here, we could write as four xy, that one's actually, that color's hard to see, let me pick a darker color. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | And so four xy is the greatest common monomial factor. And to see that, we can express each of these terms as a product of four xy and something else. So this first term right over here, so let me pick a color. So this term right over here, we could write as four xy, that one's actually, that color's hard to see, let me pick a darker color. We could write this right over here as four xy times what? And I encourage you to pause the video and think about that. Let's see, four times what is equal, is going to get us to eight? |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | So this term right over here, we could write as four xy, that one's actually, that color's hard to see, let me pick a darker color. We could write this right over here as four xy times what? And I encourage you to pause the video and think about that. Let's see, four times what is equal, is going to get us to eight? Well, four times two is going to get us to eight. X times what is going to get us to x squared? Well, x times x is going to get us to x squared. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | Let's see, four times what is equal, is going to get us to eight? Well, four times two is going to get us to eight. X times what is going to get us to x squared? Well, x times x is going to get us to x squared. And then y times what is going to get us to y? Well, it's just going to be y. So four xy times two x is actually going to give us this first term. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | Well, x times x is going to get us to x squared. And then y times what is going to get us to y? Well, it's just going to be y. So four xy times two x is actually going to give us this first term. So actually, let me just rewrite it a little bit differently. So it's four xy times two x is this first term, and you can verify that. Four times two is going to be equal to eight. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | So four xy times two x is actually going to give us this first term. So actually, let me just rewrite it a little bit differently. So it's four xy times two x is this first term, and you can verify that. Four times two is going to be equal to eight. X times x is equal to x squared. And then you just have the y. Now let's do the same thing with the second term. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | Four times two is going to be equal to eight. X times x is equal to x squared. And then you just have the y. Now let's do the same thing with the second term. And I just want to do this to show you that this is their largest common monomial factor. So the second term, and I'll do this in a slightly different color, do it in blue. I want to write this as the product of four xy and another monomial. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | Now let's do the same thing with the second term. And I just want to do this to show you that this is their largest common monomial factor. So the second term, and I'll do this in a slightly different color, do it in blue. I want to write this as the product of four xy and another monomial. So four times what is 12? Well, four times three is 12. X times what is x? |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | I want to write this as the product of four xy and another monomial. So four times what is 12? Well, four times three is 12. X times what is x? Well, it's just going to be one, so we don't have to write up anything here. And then y times what is y squared? It's going to be y times y is y squared. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | X times what is x? Well, it's just going to be one, so we don't have to write up anything here. And then y times what is y squared? It's going to be y times y is y squared. And you can verify. If you multiply these two, you're going to get 12xy squared. Four times three is 12. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | It's going to be y times y is y squared. And you can verify. If you multiply these two, you're going to get 12xy squared. Four times three is 12. You get your x. And then y times y is y squared. So so far, I've written this exact same expression, but I've taken each of those terms and I factored them into their greatest common monomial factor and then whatever is left over. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | Four times three is 12. You get your x. And then y times y is y squared. So so far, I've written this exact same expression, but I've taken each of those terms and I factored them into their greatest common monomial factor and then whatever is left over. And now I can factor the four xy out. I can actually factor it out. So this is going to be equal to, if I factor the four xys out, you could kind of say I undistribute the four xy. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | So so far, I've written this exact same expression, but I've taken each of those terms and I factored them into their greatest common monomial factor and then whatever is left over. And now I can factor the four xy out. I can actually factor it out. So this is going to be equal to, if I factor the four xys out, you could kind of say I undistribute the four xy. I factor it out. This is going to be equal to four xy times 2x plus, when I factor four xy from here, I get the three y left over. So that's three y. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | So this is going to be equal to, if I factor the four xys out, you could kind of say I undistribute the four xy. I factor it out. This is going to be equal to four xy times 2x plus, when I factor four xy from here, I get the three y left over. So that's three y. And we're done. And you can verify it. If you were to go the other way, if you were to distribute this four xy and multiply it times 2x, you'd get 8x squared y. |
Factoring binomials common factor Mathematics II High School Math Khan Academy.mp3 | So that's three y. And we're done. And you can verify it. If you were to go the other way, if you were to distribute this four xy and multiply it times 2x, you'd get 8x squared y. And then when you distribute the four xy onto the three y, you get the 12xy squared. And so we're done. This right over here is our answer. |
Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 | Well, I'm going to show you the last two logarithm properties now. So this one, and I always found this one to be in some ways the most obvious one. But don't feel bad if it's not obvious. Maybe it will take a little bit of introspection. And I encourage you to really experiment with all these logarithm properties, because that's the only way that you'll really learn them. And the point of math isn't just to pass the next exam or to get an A on the next exam. The point of math is to understand math. |
Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 | Maybe it will take a little bit of introspection. And I encourage you to really experiment with all these logarithm properties, because that's the only way that you'll really learn them. And the point of math isn't just to pass the next exam or to get an A on the next exam. The point of math is to understand math. And so you can actually apply it in life later on and not have to relearn everything every time. So the next logarithm property is if I have A times the logarithm base b of c. If I have A times this whole thing, that that equals logarithm base b of c to the a power. Fascinating. |
Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 | The point of math is to understand math. And so you can actually apply it in life later on and not have to relearn everything every time. So the next logarithm property is if I have A times the logarithm base b of c. If I have A times this whole thing, that that equals logarithm base b of c to the a power. Fascinating. So let's see if this works out. So let's say if I have, I don't know, 3 times logarithm base 2 of 8. So this property tells us that this is going to be the same thing as logarithm base 2 of 8 to the third power. |
Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 | Fascinating. So let's see if this works out. So let's say if I have, I don't know, 3 times logarithm base 2 of 8. So this property tells us that this is going to be the same thing as logarithm base 2 of 8 to the third power. And that's the same thing as, well, we could figure it out. So let's see what this is. 3 times log base, what's log base 2 of 8? |
Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 | So this property tells us that this is going to be the same thing as logarithm base 2 of 8 to the third power. And that's the same thing as, well, we could figure it out. So let's see what this is. 3 times log base, what's log base 2 of 8? The reason why I kind of hesitated a second ago is because every time I want to figure something out, I implicitly want to use log and exponential rules kind of to do it. So I'm trying to avoid that. But anyway, going back. |
Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 | 3 times log base, what's log base 2 of 8? The reason why I kind of hesitated a second ago is because every time I want to figure something out, I implicitly want to use log and exponential rules kind of to do it. So I'm trying to avoid that. But anyway, going back. So what is this? 2 to what power is 8? Well, 2 to the third power is 8, right? |
Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 | But anyway, going back. So what is this? 2 to what power is 8? Well, 2 to the third power is 8, right? So that's 3. And we have this 3 here. So 3 times 3. |
Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 | Well, 2 to the third power is 8, right? So that's 3. And we have this 3 here. So 3 times 3. So this thing right here should equal 9. If this equals 9, then we know that this property works at least for this example. You don't know if it works for all examples. |
Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 | So 3 times 3. So this thing right here should equal 9. If this equals 9, then we know that this property works at least for this example. You don't know if it works for all examples. And for that, maybe you'd want to look at the proof we have in the other videos. But that's kind of a more advanced topic. But the important thing first is just to understand how to use it. |
Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 | You don't know if it works for all examples. And for that, maybe you'd want to look at the proof we have in the other videos. But that's kind of a more advanced topic. But the important thing first is just to understand how to use it. So let's see. What is 2 to the ninth power? Well, it's going to be some large number. |
Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 | But the important thing first is just to understand how to use it. So let's see. What is 2 to the ninth power? Well, it's going to be some large number. Actually, I know what it is. It's 256, because in the last video, we figured out that 2 to the eighth was equal to 256. And so 2 to the ninth should be 512. |