I have a good understanding of the inverse functions and their domains, so I just don't get why you can even do arccos of 5.

X

I have the problem sec(arccos 5) and I understand that sec is the reciprocal of cos. But if the domain of arccos is [0,pi] then how can this be 1/5?

Relevant page
<a href="https://www.wolframalpha.com/">Wolfram|Alpha: Computational Knowledge Engine</a>
What I've done so far
I have a good understanding of the inverse functions and their domains, so I just don't get why you can even do arccos of 5.

Hi ShayT: I share your view! There is something odd about this.

A corrections first. The domain of `y=arccos(x)` is actually `-1 <=x <= 1, x ne 0.` The range however, is `0 <= y <= pi.`

Here's the graph:

But either way, we shouldn't be able to evaluate `sec(arccos 5)` since `arccos 5` is not defined. (For those not sure how this works, see: Composition of functions.)

Here's the graph of `y=sec(arccos(x))`. It's actually the same as `y=1/x` for `-1 <=x <= 1:`

Wolfram|Alpha's domain solution correctly states the domain for `sec(arccos x)`, but then goes on to plot it for -5.1 to 5.1.

They also give `sec(arccos 5) = 1/5` as per your question. I've left feedback on this issue on the W|A site and we'll see what they say.

X

Hi ShayT: I share your view! There is something odd about this.
A corrections first. The domain of `y=arccos(x)` is actually `-1 <=x <= 1, x ne 0.` The <b>range</b> however, is `0 <= y <= pi.`
Here's the graph:
[graph]310,250;-3,3;-1,5,1,1;arccos(x),[/graph]
But either way, we shouldn't be able to evaluate `sec(arccos 5)` since `arccos 5` is not defined. (For those not sure how this works, see: <a href="http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-composite-2009-1.pdf">Composition of functions</a>.)
Here's the graph of `y=sec(arccos(x))`. It's actually the same as `y=1/x` for `-1 <=x <= 1:`
[graph]310,250;-3,3;-3,3,1,1;1/cos(arccos(x)),[/graph]
<a href="http://tiny.cc/uktyiy">Wolfram|Alpha's domain solution</a> correctly states the domain for `sec(arccos x)`, but then goes on to plot it for -5.1 to 5.1.
They also give `sec(arccos 5) = 1/5` as per your question. I've left feedback on this issue on the W|A site and we'll see what they say.