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Sal introduces arccosine, which is the inverse function of cosine, and discusses its principal range. Created by Sal Khan.
Video transcript
I've already made videos on the arcsine and the arctangent, so to kind of complete the trifecta, I might as well make a video on the arccosine. And just like the other inverse trigonometric functions, the arccosine is kind of the same thought process. If I were to tell you the arc, no, I'm doing cosine, if our tell you that arccosine 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 that same statement up here. Either of these should boil down to this. If I say, you know, 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, no, two c's there, I'm told to evaluate the arccosine of minus 1/2. My brain, you know, let's say that this is going to be equal to, it's going to be equal to some angle. And this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2. 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 if I can draw a little straighter. Maybe I could actually draw, 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. Let me draw my unit circle. Looks more like a unit ellipse, but you get the idea. And the cosine of an angle as 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 a minus 1/2. So we got a minus 1/2 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 1/2. 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 this is minus 1/2 right here. Let's figure out these different angles. And the way I like to think about it is, I 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, let me do it like this. That triangle looks something like this. Where this distance right here is 1/2. That distance right there is 1/2. 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. Actually, let me just do that. Let me just call this, I don't know, just call this a. So you'd get a squared, plus 1/2 squared, which is 1/4, which is equal to 1 squared, which is 1. You get a squared is equal to 3/4, or a is equal to the square root of 3 over 2. So you immediately know this is a 30, 60, 90 triangle. And you know that because the sides of a 30, 60, 90 triangle, if the hypotenuse is 1, are 1/2 and square root of 3 over 2. And you also know that the side opposite the square root of 3 over 2 side is 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. 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. So the arccosine, or the inverse cosine, let me write that down. The arccosine of minus 1/2 is equal to 120 degrees. Did I write 180 there? 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. Or, if we wanted to write that in radians, you just right 120 degrees times pi radian per 180 degrees, degrees cancel out. 12 over 18 is 2/3, so it equals 2 pi over 3 radians. So this right here is equal to 2 pi over 3 radians. Now, just like we saw in the arcsine and the arctangent videos, you probably say, hey, OK, if I have 2 pi over 3 radians, that gives me a cosine of minus 1/2. And I can write that. cosine of 2 pi over 3 is equal to minus 1/2. This gives you the same information as this statement up here. But I can just keep going around the unit circle. For example, I could, how about this point over here? Cosine of this angle, if I were to add, if I were to go this far, would also be minus 1/2. 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 1/2. So we have to restrict ourselves. We have to restrict the values that the arccosine function can take on. So we're essentially restricting it's range. We're restricting it's range. What we do is we restrict it's range to this upper hemisphere, the first and second quadrants. So if we say, if we make the statement that the arccosine 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 102 pi. Less, oh sorry, not 2 pi. Less than or equal to pi, right? 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 minus 1/2. 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 arccosine 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 go check our work. Let's see if the value I got here, that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the TI-85. We turn it on. So i need to figure out the inverse cosine, which is the same thing as the arccosine of minus 1/2, 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 doesn't, 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, 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, you know, say I were to take the arccosine 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 can be said, well, let's say that the arccosine of x is equal to theta, that means that the cosine of theta is equal to x, right? So if the arccosine 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 get confuse you there, right? I'm saying look, arccosine of x, 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. 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 ask you what the arccosine 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, if theta is in the range, if theta is between, if theta is between 0 and pi, so it's in our valid a range for, kind of, our range for the product of the arccosine, 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. Let's take, so let me do one with theta in that range. Let's take the arccosine of the cosine of, let's just do one of them that we know. Let's take the cosine of, let's stick with cosine of 2 pi over 3. Cosine of 2 pi over 3 radians, that's the same thing as the arccosine of minus 1/2. Cosine of 2 pi over 3 is minus 1/2. We just saw that in the earlier part of this video. And then we solved this. We said, oh, this is equal to 1 pi over 3. So for in the range of thetas between 0 and pi it worked. And that's because the arccosine function can only produce values between 0 and pi. But what if I were to ask you, what is the arccosine of the cosine of, I don't know, 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 1/2 times the unit circle. So this is 3 pi. What's the x-coordinate here? It's minus 1. So cosine of 3 pi is minus 1, right? So what's arccosine of minus 1? Arccosine of minus 1. Well remember, the range, or the set of values, that arccosine can evaluate to is in this upper hemisphere. It's between, this can only be between pi and 0. So arccosine of negative 1 is just going to be pi. So this is going to be pi. Arccosine of negative, this is negative 1, arccosine 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, it's kind of, you're at the 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 arccosine of x is always going to be x. I could also do that with sine. The sine of the arcsine of x is also going to be x. And these are just useful things to, you shouldn't just memorize them, because obviously you might memorize it the wrong way, but you should just think a little bit about it, and you'll never forget It.