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# The Inverse Cosine Function - Problem 3

###### 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.

I want to graph a transformation of the inverse cosine function. I have a problem that asks me to graph y equals negative inverse cosine x plus pi over 2. I’m going to go with my usual method. I’m going to make a table of key points of inverse cosine and transform the points first.

Now I have my graph of cosine here in blue and I’ve jotted down the key points, -1 pi, zero pi over 2 and 1 zero and all I need to do is figure out what kind of transformation negative inverse cosine of x plus pi over 2 represents. Let me put that in here, negative inverse cosine of x plus pi over 2.

This minus sign just means that I’m going to be reflecting the graph across the x axis and we accomplish that by multiplying the y values here by -1. This plus pi over 2 means the graph’s going to get shifted up by pi over 2 and I do that by adding pi over 2 to the result. There’s really no horizontal transformation so I don’t have to do anything with the x coordinates so I can just write them down, -1, 0 and 1, right from here.

Let’s get ready to transform these points. We multiply by -1 and add pi over 2. We get negative pi plus pi over 2, negative pi over 2, times -1 plus pi over 2 is zero, times -1 plus pi over 2 is pi over 2. And so we have 3 points that we can put on our graph for the negative inverse cosine of x plus pi over 2. We’ve got -1, negative pi over 2, 0, 0 and 1, pi over 2. We want to draw this with the inverse cosine shape. Looks something like this.

This is the graph of y equals negative inverse cosine x plus pi over 2 but you may notice that this is also the graph of inverse sine. Y equals inverse sine of x. It turns out that this is an identity. Inverse sine of x equals negative inverse cosine of x plus pi over 2. This identity is actually related to the co-function identity.

Again whenever we graph transformations key points of the original parent graph, transform the points and then plot the points in your graph and draw a smooth curve.

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###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

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