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. So with that said, let's try it out on an example. That's 60, this is 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? Looks more like a unit ellipse, but you get the idea. The cosine of the arccosine of x is always going to be x.

Let's see if that's the same thing as 2 pi over 3. 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. And that's because the arccosine function can only produce values between 0 and pi. Let's take, so let me do one with theta in that range. So let me make this triangle a little bit bigger. This angle right here we just figured out is 60 degrees. That's 60, this is And if I know that angle, I can just subtract that from degrees to get this light blue angle that's kind of the solution to our problem. So that's my, let me see if I can draw a little straighter. So let me see, this is the angle that we're trying to figure out. And just like the other inverse trigonometric functions, the arccosine is kind of the same thought process. But this is the one we care about. And what are the valid values for x? What's the x-coordinate here? Let's take the cosine of, let's stick with cosine of 2 pi over 3. But I can just keep going around the unit circle. And to solve for that other side you just need to do the Pythagorean theorem. The cosine of the arccosine of x is always going to be x. And that's my axes. And the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle. But what if we take some theta out of that range? We said, oh, this is equal to 1 pi over 3. And the way I like to think about it is, I like to figure out this angle right here. And so you get an equivalent, it's kind of, you're at the equivalent point on the unit circle. Less than or equal to pi, right?

### Video about arcosine:

## "ASÍ LO INSULTE AL ALEMÁN" Diego Arcos - Al final del Túnel

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