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The Inverse Cosine Function - Concept
Since cosine is not a one-to-one function, the domain must be limited to 0 to pi, which is called the restricted cosine function. The inverse cosine function is written as cos^-1(x) or arccos(x). Inverse functions swap x- and y-values, so the range of inverse cosine is 0 to pi and the domain is -1 to 1. When evaluating problems, use identities or start from the inside function.
I want to talk about the inverse cosine function. We start with the function y equals cosine x I have a graph here and you can see that y equals cosine x is very much not a 1 to 1 function and we can only find the inverses of 1 to 1 functions. So we have to restrict the domain of the cosine function and the convention is to restrict it to this interval from 0 to pi so let me draw the restricted cosine function. Just this piece of the cosine graph up to and including pi and down to and including 0. So y equals cosine x for x between 0 and pi that's the restricted cosine function it is 1 to 1 and so we can invert it.
And we call this inverse y equals inverse cosine of x that's how this is read this superscript negative 1 is not an exponent it means the inverse of cosine and this function is also called y equals arc cosine x. Now I want to graph our cosine or inverse cosine and so I start with key points of the cosine curve. I've got 0, 1 pi over 2, 0 and pi negative 1, these are these 3 key points and remember when you're graphing an inverse function you just interchange the x and y coordinates so the point 0, 1 becomes 1, 0, the point pi over 2, 0 becomes 0 pi over 2 and the point pi negative 1 becomes negative 1 pi and that's going to be somewhere here. Let me connect these, keeping that the graph of a function and it's inverse have to be symmetric about the line y=x so this is a pretty good graph.
Now very important the domain, I'll mark negative 1 here, the domain of the inverse cosine function is between negative 1 and 1 very important. And think about that the cosine function can only output numbers between negative 1 and 1 so it makes sense that the domain of the inverse cosine function is this interval and the range is going to be between 0 and pi because that was the domain of the restricted cosine function and that's it. This is the graph of the inverse cosine domain between negative 1 and 1, range between 0 and pi and it has these 3 key points.