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We're told here are the approximate ratios for angle measures 25 degrees, 35 degrees, and 45 degrees. So what they're saying here is if you were to take the adjacent leg length over the hypotenuse leg length for 25 degree angle, it would be a ratio of approximately 0.91. For a 35 degree angle, it would be a ratio of 0.82, and then they do this for 45 degrees, and they do the different ratios right over here. So we're gonna use the table to approximate the measure of angle D in the triangle below. So pause this video and see if you can figure that out. All right, now let's work through this together. Now what information do they give us about angle D in this triangle? | Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3 |
So we're gonna use the table to approximate the measure of angle D in the triangle below. So pause this video and see if you can figure that out. All right, now let's work through this together. Now what information do they give us about angle D in this triangle? Well, we are given the opposite length right over here. Let me label that. That is the opposite leg length, which is 3.4. | Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3 |
Now what information do they give us about angle D in this triangle? Well, we are given the opposite length right over here. Let me label that. That is the opposite leg length, which is 3.4. We're also given, what is this right over here? Is this adjacent or is this a hypotenuse? You might be tempted to say, well, this is right next to the angle, or this is one of the lines, or it's on the ray that helps form the angle, so maybe it's adjacent. | Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3 |
That is the opposite leg length, which is 3.4. We're also given, what is this right over here? Is this adjacent or is this a hypotenuse? You might be tempted to say, well, this is right next to the angle, or this is one of the lines, or it's on the ray that helps form the angle, so maybe it's adjacent. But remember, adjacent is the adjacent side that is not the hypotenuse. And this is clearly the hypotenuse. It is the longest side. | Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3 |
You might be tempted to say, well, this is right next to the angle, or this is one of the lines, or it's on the ray that helps form the angle, so maybe it's adjacent. But remember, adjacent is the adjacent side that is not the hypotenuse. And this is clearly the hypotenuse. It is the longest side. It is the side opposite the 90 degree angle. So this right over here is the hypotenuse. Hypotenuse. | Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3 |
It is the longest side. It is the side opposite the 90 degree angle. So this right over here is the hypotenuse. Hypotenuse. So we're given the opposite leg length and the hypotenuse length. And so let's see, which of these ratios deal with the opposite and the hypotenuse? And if we, let's see, this first one is adjacent and hypotenuse. | Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3 |
Hypotenuse. So we're given the opposite leg length and the hypotenuse length. And so let's see, which of these ratios deal with the opposite and the hypotenuse? And if we, let's see, this first one is adjacent and hypotenuse. The second one here is hypotenuse, sorry, opposite and hypotenuse. So that's exactly what we're talking about. We're talking about the opposite leg length over the hypotenuse, over the hypotenuse length. | Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3 |
And if we, let's see, this first one is adjacent and hypotenuse. The second one here is hypotenuse, sorry, opposite and hypotenuse. So that's exactly what we're talking about. We're talking about the opposite leg length over the hypotenuse, over the hypotenuse length. So in this case, what is going to be our opposite leg length over our hypotenuse leg length? It's going to be 3.4 over eight. 3.4 over eight, which is approximately going to be equal to, let me do this down here, this is eight goes into 3.4. | Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3 |
We're talking about the opposite leg length over the hypotenuse, over the hypotenuse length. So in this case, what is going to be our opposite leg length over our hypotenuse leg length? It's going to be 3.4 over eight. 3.4 over eight, which is approximately going to be equal to, let me do this down here, this is eight goes into 3.4. Eight goes in, doesn't go into three. Eight goes into 34 four times. Four times eight is 32. | Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3 |
3.4 over eight, which is approximately going to be equal to, let me do this down here, this is eight goes into 3.4. Eight goes in, doesn't go into three. Eight goes into 34 four times. Four times eight is 32. If I subtract, and I could scroll down a little bit, I get a two. I can bring down a zero. Eight goes into 20 two times. | Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3 |
Four times eight is 32. If I subtract, and I could scroll down a little bit, I get a two. I can bring down a zero. Eight goes into 20 two times. And that's about as much precision as any of these have. And so it looks like for this particular triangle and this angle of the triangle, if I were to take a ratio of the opposite length and the hypotenuse length, opposite over hypotenuse, I get 0.42. So that looks like this situation right over here. | Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3 |
We use it in everyday language. We've done some examples on this playlist where if you had an angle like that, you might call that a 30-degree angle. If you have an angle like this, you could call that a 90-degree angle, and we'd often use this symbol just like that. If you were to go 180 degrees, you'd essentially form a straight line. Let me make these proper angles. If you go 360 degrees, you've essentially done one full rotation. If you watch figure skating on the Olympics and someone does a rotation, they'll say, oh, they did a 360, or especially in some skateboarding competitions and things like that. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
If you were to go 180 degrees, you'd essentially form a straight line. Let me make these proper angles. If you go 360 degrees, you've essentially done one full rotation. If you watch figure skating on the Olympics and someone does a rotation, they'll say, oh, they did a 360, or especially in some skateboarding competitions and things like that. But the one thing to realize, and it might not be obvious right from the get-go, is this whole notion of degrees, this is a human-constructed system. This is not the only way that you can measure angles. If you think about it, you say, well, why do we call a full rotation 360 degrees? | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
If you watch figure skating on the Olympics and someone does a rotation, they'll say, oh, they did a 360, or especially in some skateboarding competitions and things like that. But the one thing to realize, and it might not be obvious right from the get-go, is this whole notion of degrees, this is a human-constructed system. This is not the only way that you can measure angles. If you think about it, you say, well, why do we call a full rotation 360 degrees? There are some possible theories, and I encourage you to think about them. Why does 360 degrees show up in our culture as a full rotation? Well, there's a couple of theories there. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
If you think about it, you say, well, why do we call a full rotation 360 degrees? There are some possible theories, and I encourage you to think about them. Why does 360 degrees show up in our culture as a full rotation? Well, there's a couple of theories there. One is ancient calendars, and even our calendar is close to this, but ancient calendars were based on 360 days in a year. Some ancient astronomers observed that things seemed to move 1 360th of the sky per day. Another theory is the ancient Babylonians liked equilateral triangles a lot, and they had a base 60 number system, so they had 60 symbols. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
Well, there's a couple of theories there. One is ancient calendars, and even our calendar is close to this, but ancient calendars were based on 360 days in a year. Some ancient astronomers observed that things seemed to move 1 360th of the sky per day. Another theory is the ancient Babylonians liked equilateral triangles a lot, and they had a base 60 number system, so they had 60 symbols. We only have 10. We have a base 10. They had 60. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
Another theory is the ancient Babylonians liked equilateral triangles a lot, and they had a base 60 number system, so they had 60 symbols. We only have 10. We have a base 10. They had 60. In our system, we like to divide things into 10. They probably like to divide things into 60. If you had a circle and you divided it into 6 equilateral triangles, and each of those equilateral triangles you divided into 60 sections because you have a base 60 number system, then you might end up with 360 degrees. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
They had 60. In our system, we like to divide things into 10. They probably like to divide things into 60. If you had a circle and you divided it into 6 equilateral triangles, and each of those equilateral triangles you divided into 60 sections because you have a base 60 number system, then you might end up with 360 degrees. What I want to think about in this video is an alternate way of measuring angles. That alternate way, even though it might not seem as intuitive to you from the get-go, in some ways is much more mathematically pure than degrees. It's not based on these cultural artifacts of base 60 number systems or astronomical patterns. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
If you had a circle and you divided it into 6 equilateral triangles, and each of those equilateral triangles you divided into 60 sections because you have a base 60 number system, then you might end up with 360 degrees. What I want to think about in this video is an alternate way of measuring angles. That alternate way, even though it might not seem as intuitive to you from the get-go, in some ways is much more mathematically pure than degrees. It's not based on these cultural artifacts of base 60 number systems or astronomical patterns. To some degree, an alien on another planet would not use degrees, especially if the degrees are motivated by these astronomical phenomena, but they might use what we're going to define as a radian. There's a certain degree of purity here. Radians. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
It's not based on these cultural artifacts of base 60 number systems or astronomical patterns. To some degree, an alien on another planet would not use degrees, especially if the degrees are motivated by these astronomical phenomena, but they might use what we're going to define as a radian. There's a certain degree of purity here. Radians. Let's just cut to the chase and define what a radian is. Let me draw a circle here. My best attempt at drawing a circle. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
Radians. Let's just cut to the chase and define what a radian is. Let me draw a circle here. My best attempt at drawing a circle. Not bad. Let me draw the center of the circle. Now let me draw this radius. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
My best attempt at drawing a circle. Not bad. Let me draw the center of the circle. Now let me draw this radius. You might already notice the word radius is very close to the word radians, and that's not a coincidence. Let's say that this circle has a radius of length r. Let's construct an angle. I'll call that angle theta. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
Now let me draw this radius. You might already notice the word radius is very close to the word radians, and that's not a coincidence. Let's say that this circle has a radius of length r. Let's construct an angle. I'll call that angle theta. Let's construct an angle theta. Let's call this angle right over here theta. Let's just say, for the sake of argument, that this angle is just the exact right measure. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
I'll call that angle theta. Let's construct an angle theta. Let's call this angle right over here theta. Let's just say, for the sake of argument, that this angle is just the exact right measure. If you look at the arc that subtends this angle, that seems like a very fancy word. Let me draw the angle. If you look at the arc that subtends the angle, that's a fancy word, but that's really just talking about the arc along the circle that intersects the two sides of the angles. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
Let's just say, for the sake of argument, that this angle is just the exact right measure. If you look at the arc that subtends this angle, that seems like a very fancy word. Let me draw the angle. If you look at the arc that subtends the angle, that's a fancy word, but that's really just talking about the arc along the circle that intersects the two sides of the angles. This arc right over here subtends the angle theta. Let me write that down. Subtends this arc. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
If you look at the arc that subtends the angle, that's a fancy word, but that's really just talking about the arc along the circle that intersects the two sides of the angles. This arc right over here subtends the angle theta. Let me write that down. Subtends this arc. Subtends angle theta. Let's say theta is the exact right size. This arc is also the same length as the radius of the circle. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
Subtends this arc. Subtends angle theta. Let's say theta is the exact right size. This arc is also the same length as the radius of the circle. This arc is also of length r. Given that, if you were defining a new type of angle measurement, and you wanted to call it a radian, which is very close to a radius, how many radians would you define this angle to be? The most obvious one, if you view a radian as another way of saying radiuses or radii, you say, look, this is subtended by an arc of one radius, so why don't we call this right over here one radian? Which is exactly how a radian is defined. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
This arc is also the same length as the radius of the circle. This arc is also of length r. Given that, if you were defining a new type of angle measurement, and you wanted to call it a radian, which is very close to a radius, how many radians would you define this angle to be? The most obvious one, if you view a radian as another way of saying radiuses or radii, you say, look, this is subtended by an arc of one radius, so why don't we call this right over here one radian? Which is exactly how a radian is defined. When you have a circle and you have an angle of one radian, the arc that subtends it is exactly one radius long, which you can imagine might be a little bit useful as we start to interpret more and more types of circles. When you give a degree, you really have to do a little bit of math and think about the circumference and all of that to think about how many radiuses are subtending that angle. Here, the angle in radians tells you exactly how many arc lengths that is subtending the angle. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
Which is exactly how a radian is defined. When you have a circle and you have an angle of one radian, the arc that subtends it is exactly one radius long, which you can imagine might be a little bit useful as we start to interpret more and more types of circles. When you give a degree, you really have to do a little bit of math and think about the circumference and all of that to think about how many radiuses are subtending that angle. Here, the angle in radians tells you exactly how many arc lengths that is subtending the angle. Let's do a couple of thought experiments here. Given that, what would be the angle in radians if we were to go... Let me draw another circle here. Let me draw another circle here. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
Here, the angle in radians tells you exactly how many arc lengths that is subtending the angle. Let's do a couple of thought experiments here. Given that, what would be the angle in radians if we were to go... Let me draw another circle here. Let me draw another circle here. That's the center. We'll start right over there. What would happen if I had an angle? | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
Let me draw another circle here. That's the center. We'll start right over there. What would happen if I had an angle? What angle, if I wanted to measure in radians, what angle would this be in radians? You can almost think of it as radiuses. What would that angle be? | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
What would happen if I had an angle? What angle, if I wanted to measure in radians, what angle would this be in radians? You can almost think of it as radiuses. What would that angle be? Going one full revolution in degrees, that would be 360 degrees. Based on this definition, what would this be in radians? Let's think about the arc that subtends this angle. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
What would that angle be? Going one full revolution in degrees, that would be 360 degrees. Based on this definition, what would this be in radians? Let's think about the arc that subtends this angle. The arc that subtends this angle is the entire circumference of this circle. It's the entire circumference of this circle. What's the circumference of a circle in terms of radiuses? | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
Let's think about the arc that subtends this angle. The arc that subtends this angle is the entire circumference of this circle. It's the entire circumference of this circle. What's the circumference of a circle in terms of radiuses? If this has length r, if the radius is length r, what's the circumference of the circle in terms of r? We know that. That's going to be 2 pi r. Going back to this angle, the length of the arc that subtends this angle is how many radiuses this is? | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
What's the circumference of a circle in terms of radiuses? If this has length r, if the radius is length r, what's the circumference of the circle in terms of r? We know that. That's going to be 2 pi r. Going back to this angle, the length of the arc that subtends this angle is how many radiuses this is? What's 2 pi radiuses this is? It's 2 pi times r. This angle right over here, I'll call this a different angle, x. x in this case is going to be 2 pi radians. It is subtended by an arc length of 2 pi radiuses. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
That's going to be 2 pi r. Going back to this angle, the length of the arc that subtends this angle is how many radiuses this is? What's 2 pi radiuses this is? It's 2 pi times r. This angle right over here, I'll call this a different angle, x. x in this case is going to be 2 pi radians. It is subtended by an arc length of 2 pi radiuses. If the radius was one unit, then this would be 2 pi times 1, 2 pi radiuses. Given that, let's start to think about how we can convert between radians and degrees and vice versa. If I were to have, and we can just follow up over here, if we do one full revolution, that is 2 pi radians, how many degrees is this going to be equal to? | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
It is subtended by an arc length of 2 pi radiuses. If the radius was one unit, then this would be 2 pi times 1, 2 pi radiuses. Given that, let's start to think about how we can convert between radians and degrees and vice versa. If I were to have, and we can just follow up over here, if we do one full revolution, that is 2 pi radians, how many degrees is this going to be equal to? We already know this. A full revolution in degrees is 360 degrees. I could either write out the word degrees or I can use this little degree notation there. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
If I were to have, and we can just follow up over here, if we do one full revolution, that is 2 pi radians, how many degrees is this going to be equal to? We already know this. A full revolution in degrees is 360 degrees. I could either write out the word degrees or I can use this little degree notation there. Actually, let me write out the word degrees. It might make things a little bit clearer that we're using units in both cases. If we wanted to simplify this a little bit, we could divide both sides by 2, in which case we would get on the left-hand side, we would get pi radians would be equal to how many degrees? | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
I could either write out the word degrees or I can use this little degree notation there. Actually, let me write out the word degrees. It might make things a little bit clearer that we're using units in both cases. If we wanted to simplify this a little bit, we could divide both sides by 2, in which case we would get on the left-hand side, we would get pi radians would be equal to how many degrees? It would be equal to 180 degrees. 180 degrees. I could write it that way or I could write it that way. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
If we wanted to simplify this a little bit, we could divide both sides by 2, in which case we would get on the left-hand side, we would get pi radians would be equal to how many degrees? It would be equal to 180 degrees. 180 degrees. I could write it that way or I could write it that way. You see over here, this is 180 degrees. You also see if you were to draw a circle around here, we've gone halfway around the circle. The arc length or the arc that subtends the angle is half the circumference. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
I could write it that way or I could write it that way. You see over here, this is 180 degrees. You also see if you were to draw a circle around here, we've gone halfway around the circle. The arc length or the arc that subtends the angle is half the circumference. Half the circumference are pi radiuses. We call this pi radians. Pi radians is 180 degrees. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
The arc length or the arc that subtends the angle is half the circumference. Half the circumference are pi radiuses. We call this pi radians. Pi radians is 180 degrees. From this, we can come up with conversions. One radian would be how many degrees? To do that, we would just have to divide both sides by pi. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
Pi radians is 180 degrees. From this, we can come up with conversions. One radian would be how many degrees? To do that, we would just have to divide both sides by pi. On the left-hand side, you'd be left with 1. I'll just write it singular now. One radian is equal to, I'm just dividing both sides. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
To do that, we would just have to divide both sides by pi. On the left-hand side, you'd be left with 1. I'll just write it singular now. One radian is equal to, I'm just dividing both sides. Let me make it clear what I'm doing here just to show you. This isn't some voodoo. I'm just dividing both sides by pi here. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
One radian is equal to, I'm just dividing both sides. Let me make it clear what I'm doing here just to show you. This isn't some voodoo. I'm just dividing both sides by pi here. On the left-hand side, you're left with 1. On the right-hand side, you're left with 180 over pi degrees. One radian is equal to 180 over pi degrees, which is starting to make it an interesting way to convert them. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
I'm just dividing both sides by pi here. On the left-hand side, you're left with 1. On the right-hand side, you're left with 180 over pi degrees. One radian is equal to 180 over pi degrees, which is starting to make it an interesting way to convert them. Let's think about it the other way. If I were to have one degree, how many radians is that? Let's start off with, let me rewrite this thing over here. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
One radian is equal to 180 over pi degrees, which is starting to make it an interesting way to convert them. Let's think about it the other way. If I were to have one degree, how many radians is that? Let's start off with, let me rewrite this thing over here. We said pi radians is equal to 180 degrees. Now we want to think about one degree. Let's solve for one degree. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
Let's start off with, let me rewrite this thing over here. We said pi radians is equal to 180 degrees. Now we want to think about one degree. Let's solve for one degree. One degree, we can divide both sides by 180. We are left with pi over 180 radians is equal to one degree. Pi over 180 radians is equal to one degree. | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
Let's solve for one degree. One degree, we can divide both sides by 180. We are left with pi over 180 radians is equal to one degree. Pi over 180 radians is equal to one degree. This might seem confusing and daunting, and it was for me the first time I was exposed to it, especially because we're not exposed to this in our everyday life. What we're going to see over the next few examples is that as long as we keep in mind this whole idea that 2 pi radians is equal to 360 degrees or that pi radians is equal to 180 degrees, which is the two things that I do keep in my mind, we can always re-derive these two things. You might say, hey, how do I remember if it's pi over 180 or 180 over pi to convert the two things? | Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3 |
I've already made videos on the arc sine and the arc tangent. So to kind of complete the trifecta, I might as well make a video on the arc cosine. And just like the other inverse trigonometric functions, the arc cosine is kind of the same thought process. If I were to tell you that the arc cosine of x is equal to theta, this is an equivalent statement to saying that the inverse cosine of x is equal to theta. These are just two different ways of writing the exact same thing. And as soon as I see either an arc anything or an inverse trig function in general, my brain immediately rearranges this, my brain immediately says, this is saying that if I take the cosine of some angle theta, that I'm going to get x. Or the same statement up here, either of these should boil down to this. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
If I were to tell you that the arc cosine of x is equal to theta, this is an equivalent statement to saying that the inverse cosine of x is equal to theta. These are just two different ways of writing the exact same thing. And as soon as I see either an arc anything or an inverse trig function in general, my brain immediately rearranges this, my brain immediately says, this is saying that if I take the cosine of some angle theta, that I'm going to get x. Or the same statement up here, either of these should boil down to this. If I say, what is the inverse cosine of x, my brain says, what angle can I take the cosine of to get x? So with that said, let's try it out on an example. Let's say that I have the arc. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
Or the same statement up here, either of these should boil down to this. If I say, what is the inverse cosine of x, my brain says, what angle can I take the cosine of to get x? So with that said, let's try it out on an example. Let's say that I have the arc. I'm told to evaluate the arc cosine of minus 1 half. My brain, let's say that this is going to be equal to some angle, and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1 half. And as soon as you put it in this way, at least for my brain, it becomes a lot easier to process. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
Let's say that I have the arc. I'm told to evaluate the arc cosine of minus 1 half. My brain, let's say that this is going to be equal to some angle, and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1 half. And as soon as you put it in this way, at least for my brain, it becomes a lot easier to process. So let's draw our unit circle and see if we can make some headway here. So that's my, let me see, I could draw a little straighter. Actually, maybe I could actually put rulers here. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
And as soon as you put it in this way, at least for my brain, it becomes a lot easier to process. So let's draw our unit circle and see if we can make some headway here. So that's my, let me see, I could draw a little straighter. Actually, maybe I could actually put rulers here. And if I put a ruler here, maybe I can draw a straight line, let me see. No, that's too hard. OK, so that is my y-axis. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
Actually, maybe I could actually put rulers here. And if I put a ruler here, maybe I can draw a straight line, let me see. No, that's too hard. OK, so that is my y-axis. That is my x-axis, not the most neatly drawn axes ever, but it'll do. And let me draw my unit circle. Looks more like a unit ellipse, but you get the idea. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
OK, so that is my y-axis. That is my x-axis, not the most neatly drawn axes ever, but it'll do. And let me draw my unit circle. Looks more like a unit ellipse, but you get the idea. And the cosine of an angle, it's defined on the unit circle definition, is the x value on the unit circle. So if we have some angle, the x value is going to be equal to minus 1 half. So we go to minus 1 half right here. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
Looks more like a unit ellipse, but you get the idea. And the cosine of an angle, it's defined on the unit circle definition, is the x value on the unit circle. So if we have some angle, the x value is going to be equal to minus 1 half. So we go to minus 1 half right here. And so the angle that we have to solve for, our theta, is the angle that when we intersect the unit circle, the x value is minus 1 half. So let me see, this is the angle that we're trying to figure out. This is theta that we need to determine. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
So we go to minus 1 half right here. And so the angle that we have to solve for, our theta, is the angle that when we intersect the unit circle, the x value is minus 1 half. So let me see, this is the angle that we're trying to figure out. This is theta that we need to determine. So how can we do that? So if this is minus 1 half right here, let's figure out these different angles. And the way I like to think about it, and it's not like to figure out this angle right here. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
This is theta that we need to determine. So how can we do that? So if this is minus 1 half right here, let's figure out these different angles. And the way I like to think about it, and it's not like to figure out this angle right here. And if I know that angle, I can just subtract that from 180 degrees to get this light blue angle that's kind of the solution to our problem. So let me make this triangle a little bit bigger. So that triangle looks something like this, where this distance right here is 1 half. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
And the way I like to think about it, and it's not like to figure out this angle right here. And if I know that angle, I can just subtract that from 180 degrees to get this light blue angle that's kind of the solution to our problem. So let me make this triangle a little bit bigger. So that triangle looks something like this, where this distance right here is 1 half. That distance right there is 1 half. This distance right here is 1. Hopefully you recognize that this is going to be a 30-60-90 triangle. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
So that triangle looks something like this, where this distance right here is 1 half. That distance right there is 1 half. This distance right here is 1. Hopefully you recognize that this is going to be a 30-60-90 triangle. You could actually solve for this other side. You'll get the square root of 3 over 2. And to solve for that other side, you just need to do the Pythagorean theorem. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
Hopefully you recognize that this is going to be a 30-60-90 triangle. You could actually solve for this other side. You'll get the square root of 3 over 2. And to solve for that other side, you just need to do the Pythagorean theorem. Actually, let me just do that. Let me just call this a. So you'd get a squared plus 1 half squared, which is 1 fourth, is equal to 1 squared, which is 1. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
And to solve for that other side, you just need to do the Pythagorean theorem. Actually, let me just do that. Let me just call this a. So you'd get a squared plus 1 half squared, which is 1 fourth, is equal to 1 squared, which is 1. You'd get a squared is equal to 3 fourths, or a is equal to the square root of 3 over 2. So you immediately notice it's a 30-60-90 triangle. And you know that because the sides of a 30-60-90 triangle, if the hypotenuse is 1, are 1 half and square root of 3 over 2. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
So you'd get a squared plus 1 half squared, which is 1 fourth, is equal to 1 squared, which is 1. You'd get a squared is equal to 3 fourths, or a is equal to the square root of 3 over 2. So you immediately notice it's a 30-60-90 triangle. And you know that because the sides of a 30-60-90 triangle, if the hypotenuse is 1, are 1 half and square root of 3 over 2. And you'll also know that the side opposite the square root of 3 over 2 side is the 60 degrees. That's 60. This is 90. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
And you know that because the sides of a 30-60-90 triangle, if the hypotenuse is 1, are 1 half and square root of 3 over 2. And you'll also know that the side opposite the square root of 3 over 2 side is the 60 degrees. That's 60. This is 90. This is the right angle. And this is 30 right up there. But this is the one we care about. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
This is 90. This is the right angle. And this is 30 right up there. But this is the one we care about. This angle right here. We just figured out is 60 degrees. So what's this? | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
But this is the one we care about. This angle right here. We just figured out is 60 degrees. So what's this? What's the bigger angle that we care about? What is 60 degrees supplementary to? It's supplementary to 180 degrees. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
So what's this? What's the bigger angle that we care about? What is 60 degrees supplementary to? It's supplementary to 180 degrees. So the arc cosine, or the inverse cosine, let me write that down, the arc cosine of minus 1 half is equal to 120 degrees. No, it's 180 minus the 60. This whole thing is 180. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
It's supplementary to 180 degrees. So the arc cosine, or the inverse cosine, let me write that down, the arc cosine of minus 1 half is equal to 120 degrees. No, it's 180 minus the 60. This whole thing is 180. So this is right here is 120 degrees. Right? 120 plus 60 is 180. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
This whole thing is 180. So this is right here is 120 degrees. Right? 120 plus 60 is 180. Or if we wanted to write that in radians, you just write 120 degrees times pi radian per 180 degrees. Degrees cancel out. 12 over 18 is 2 thirds. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
120 plus 60 is 180. Or if we wanted to write that in radians, you just write 120 degrees times pi radian per 180 degrees. Degrees cancel out. 12 over 18 is 2 thirds. So it equals 2 pi over 3 radians. So this right here is equal to 2 pi pi over 3 radians. Now, just like we saw in the arc sine and the arc tangent videos, you probably say, hey, OK, if I have 2 pi over 3 radians, that gives me a cosine of minus 1 half. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
12 over 18 is 2 thirds. So it equals 2 pi over 3 radians. So this right here is equal to 2 pi pi over 3 radians. Now, just like we saw in the arc sine and the arc tangent videos, you probably say, hey, OK, if I have 2 pi over 3 radians, that gives me a cosine of minus 1 half. And I could write that. Cosine of 2 pi over 3 is equal to minus 1 half. This gives you the same information as this statement up here. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
Now, just like we saw in the arc sine and the arc tangent videos, you probably say, hey, OK, if I have 2 pi over 3 radians, that gives me a cosine of minus 1 half. And I could write that. Cosine of 2 pi over 3 is equal to minus 1 half. This gives you the same information as this statement up here. But I could just keep going around the unit circle. For example, I could, well, this point over here, cosine of this angle, if I were to go this far, would also be minus 1 half. And then I could go 2 pi around and get back here. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
This gives you the same information as this statement up here. But I could just keep going around the unit circle. For example, I could, well, this point over here, cosine of this angle, if I were to go this far, would also be minus 1 half. And then I could go 2 pi around and get back here. So there's a lot of values that if I take the cosine of those angles, I'll get this minus 1 half. So we have to restrict ourselves. We have to restrict the values that the arc cosine function can take on. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
And then I could go 2 pi around and get back here. So there's a lot of values that if I take the cosine of those angles, I'll get this minus 1 half. So we have to restrict ourselves. We have to restrict the values that the arc cosine function can take on. So we're essentially restricting its range. What we do is we restrict the range to this upper hemisphere, the first and second quadrants. So if we make the statement that the arc cosine of x is equal to theta, we're going to restrict our range, theta, to that top. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
We have to restrict the values that the arc cosine function can take on. So we're essentially restricting its range. What we do is we restrict the range to this upper hemisphere, the first and second quadrants. So if we make the statement that the arc cosine of x is equal to theta, we're going to restrict our range, theta, to that top. So theta is going to be greater than or equal to 0 and less than or equal to pi, where this is also 0 degrees or 180 degrees. We're restricting ourselves to this part of the hemisphere right there. And so you can't do this. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
So if we make the statement that the arc cosine of x is equal to theta, we're going to restrict our range, theta, to that top. So theta is going to be greater than or equal to 0 and less than or equal to pi, where this is also 0 degrees or 180 degrees. We're restricting ourselves to this part of the hemisphere right there. And so you can't do this. This is the only point where the cosine of the angle is equal to minus 1 half. We can't take this angle because it's outside of our range. And what are the valid values for x? | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
And so you can't do this. This is the only point where the cosine of the angle is equal to minus 1 half. We can't take this angle because it's outside of our range. And what are the valid values for x? Well, any angle, if I take the cosine of it, it can be between minus 1 and plus 1. So x, the domain for the arc cosine function, is going to be x has to be less than or equal to 1 and greater than or equal to minus 1. And once again, let's just check our work. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
And what are the valid values for x? Well, any angle, if I take the cosine of it, it can be between minus 1 and plus 1. So x, the domain for the arc cosine function, is going to be x has to be less than or equal to 1 and greater than or equal to minus 1. And once again, let's just check our work. Let's see if the value I got here, that the arc cosine of minus 1 half, really is 2 pi over 3, as calculated by the TI-85. Let me turn it on. So I need to figure out the inverse cosine, which is the same thing as the arc cosine, of minus 1 half, of minus 0.5. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
And once again, let's just check our work. Let's see if the value I got here, that the arc cosine of minus 1 half, really is 2 pi over 3, as calculated by the TI-85. Let me turn it on. So I need to figure out the inverse cosine, which is the same thing as the arc cosine, of minus 1 half, of minus 0.5. It gives me that decimal, that strange number. Let's see if that's the same thing as 2 pi over 3. 2 times pi divided by 3 is equal to that exact same number, so the calculator gave me the same value I got. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
So I need to figure out the inverse cosine, which is the same thing as the arc cosine, of minus 1 half, of minus 0.5. It gives me that decimal, that strange number. Let's see if that's the same thing as 2 pi over 3. 2 times pi divided by 3 is equal to that exact same number, so the calculator gave me the same value I got. But this is kind of a useless, well, it's not a useless number, it's a valid, that is the answer. But it's not a nice, clean answer. I didn't know that this is 2 pi over 3 radians. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
2 times pi divided by 3 is equal to that exact same number, so the calculator gave me the same value I got. But this is kind of a useless, well, it's not a useless number, it's a valid, that is the answer. But it's not a nice, clean answer. I didn't know that this is 2 pi over 3 radians. And so when we did it using the unit circle, we were able to get that answer. So hopefully, and actually, let me ask you, let me just finish this up with an interesting question. And this applies to all of them. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
I didn't know that this is 2 pi over 3 radians. And so when we did it using the unit circle, we were able to get that answer. So hopefully, and actually, let me ask you, let me just finish this up with an interesting question. And this applies to all of them. If I were to ask you, say I were to take the arc cosine of x, and then I were to take the cosine of that, what is this going to be equal to? Well, this statement right here could be said, well, let's say that the arc cosine of x is equal to theta. That means that the cosine of theta is equal to x. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
And this applies to all of them. If I were to ask you, say I were to take the arc cosine of x, and then I were to take the cosine of that, what is this going to be equal to? Well, this statement right here could be said, well, let's say that the arc cosine of x is equal to theta. That means that the cosine of theta is equal to x. So if the arc cosine of x is equal to theta, we can replace this with theta. And then the cosine of theta, well, the cosine of theta is x. So this whole thing is going to be x. Hopefully, I didn't confuse you there. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
That means that the cosine of theta is equal to x. So if the arc cosine of x is equal to theta, we can replace this with theta. And then the cosine of theta, well, the cosine of theta is x. So this whole thing is going to be x. Hopefully, I didn't confuse you there. I'm just saying, look, arc cosine of x, let's just call that theta. Now, by definition, this means that the cosine of theta is equal to x. These are equivalent statements. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
So this whole thing is going to be x. Hopefully, I didn't confuse you there. I'm just saying, look, arc cosine of x, let's just call that theta. Now, by definition, this means that the cosine of theta is equal to x. These are equivalent statements. These are completely equivalent statements right here. So if we put a theta right there, we take the cosine of theta, it has to be equal to x. Now, let me ask you a bonus, slightly trickier question. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
These are equivalent statements. These are completely equivalent statements right here. So if we put a theta right there, we take the cosine of theta, it has to be equal to x. Now, let me ask you a bonus, slightly trickier question. What if I were to ask you, and this is true for any x that you put in here, this is true for any x, any value between negative 1 and 1, including those two endpoints. This is going to be true. Now, what if I were to ask you what the arc cosine of the cosine of theta is? | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
Now, let me ask you a bonus, slightly trickier question. What if I were to ask you, and this is true for any x that you put in here, this is true for any x, any value between negative 1 and 1, including those two endpoints. This is going to be true. Now, what if I were to ask you what the arc cosine of the cosine of theta is? What is this going to be equal to? My answer is, it depends on the theta. So if theta is in the range, if theta is between 0 and pi, so it's in our valid range for the product of the arc cosine, then this will be equal to theta, if this is true for theta. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
Now, what if I were to ask you what the arc cosine of the cosine of theta is? What is this going to be equal to? My answer is, it depends on the theta. So if theta is in the range, if theta is between 0 and pi, so it's in our valid range for the product of the arc cosine, then this will be equal to theta, if this is true for theta. But what if we take some theta out of that range? Let's try it out. So let me do it one with theta in that range. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
So if theta is in the range, if theta is between 0 and pi, so it's in our valid range for the product of the arc cosine, then this will be equal to theta, if this is true for theta. But what if we take some theta out of that range? Let's try it out. So let me do it one with theta in that range. Let's take the arc cosine of the cosine of, let's just do some one of them that we know, let's take the cosine of 2 pi over 3 radians. That's the same thing as the arc cosine of minus 1 half. Cosine of 2 pi over 3 is minus 1 half. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
So let me do it one with theta in that range. Let's take the arc cosine of the cosine of, let's just do some one of them that we know, let's take the cosine of 2 pi over 3 radians. That's the same thing as the arc cosine of minus 1 half. Cosine of 2 pi over 3 is minus 1 half. We just saw that in the earlier part of this video. And then we solved this. We said, oh, this is equal to 2 pi over 3. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
Cosine of 2 pi over 3 is minus 1 half. We just saw that in the earlier part of this video. And then we solved this. We said, oh, this is equal to 2 pi over 3. So if we're in the range, if theta is between 0 and pi, it worked. And that's because the arc cosine function can only produce values between 0 and pi. But what if I were to ask you, what is the arc cosine of 3 pi? | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
We said, oh, this is equal to 2 pi over 3. So if we're in the range, if theta is between 0 and pi, it worked. And that's because the arc cosine function can only produce values between 0 and pi. But what if I were to ask you, what is the arc cosine of 3 pi? So if I were to draw the unit circle here, let me draw the unit circle, a real quick one. And that's my axes. What's 3 pi? | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
But what if I were to ask you, what is the arc cosine of 3 pi? So if I were to draw the unit circle here, let me draw the unit circle, a real quick one. And that's my axes. What's 3 pi? 2 pi is if I go around once, and then I go around another pi. So I end up right here. So I've gone around 1 and 1 half times the unit circle. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
What's 3 pi? 2 pi is if I go around once, and then I go around another pi. So I end up right here. So I've gone around 1 and 1 half times the unit circle. So is it 3 pi? What's the x-coordinate here? It's minus 1. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
So I've gone around 1 and 1 half times the unit circle. So is it 3 pi? What's the x-coordinate here? It's minus 1. So cosine of 3 pi is minus 1. So what's arc cosine of minus 1? Arc cosine of minus 1. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
It's minus 1. So cosine of 3 pi is minus 1. So what's arc cosine of minus 1? Arc cosine of minus 1. Well, remember, the range or the set of values that arc cosine can evaluate to is in this upper hemisphere. This can only be between pi and 0. So arc cosine of negative 1 is just going to be pi. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
Arc cosine of minus 1. Well, remember, the range or the set of values that arc cosine can evaluate to is in this upper hemisphere. This can only be between pi and 0. So arc cosine of negative 1 is just going to be pi. So this is going to be pi. Arc cosine of negative 1 is pi. And that's a reasonable statement, because the difference between 3 pi and pi is just going around the unit circle a couple of times. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
So arc cosine of negative 1 is just going to be pi. So this is going to be pi. Arc cosine of negative 1 is pi. And that's a reasonable statement, because the difference between 3 pi and pi is just going around the unit circle a couple of times. And so you get an equivalent point on the unit circle. So I just thought I would throw those two at you. This one, I mean, this is a useful one. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
And that's a reasonable statement, because the difference between 3 pi and pi is just going around the unit circle a couple of times. And so you get an equivalent point on the unit circle. So I just thought I would throw those two at you. This one, I mean, this is a useful one. Well, actually, let me write it up here. This one is a useful one. The cosine of the arc cosine of x is always going to be x. I could also do that with sine. | Inverse trig functions arccos Trigonometry Khan Academy.mp3 |
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