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The Inverse Cosine Function - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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.

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