# Where does cosh(ix) disppaer to?

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For arccos(x), outside of its domain on [-1,1], it turns into i*arccosh(x). But, if you took the inverse of accros(x) (no h), those parabola-like structures caused by i*cosh(x) disappear when that function is converted into the function cos(x). I want to know why that I*cosh(x) component suddenly disappears when I flip arccos(x) over the x axis to find the inverse, or if they are still there but just not being displayed?

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It sounds like a limitation of the display software.

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Well I can't say that for sure because when studying hyperbolic functions, their relation to the cos(x) graph was never mentioned, so I don't know if its purposely not showing me that imaginary component of if imaginary components sometimes stop existing when you take the inverse a function.

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• 1 month later...

Wow I can't believe no one familiar with math is answered this, it's definitely not some millennium \$1,000,000 problem, it has a real answer. Where do the imaginary hyperbolas go when I take the inverse of arccos(x) (that's right, ARCcos(x), NOT cos(x)).?

See? Past -1 and 1, the equation turns into something like -i*arccosh(x)+pi. You can very very very very very very clearly see those red, imaginary logarithms or hyperbolas and see the clear blue arccos(x) on [-1,1] that we're all familiar with. BUT, now look what happens when I take the inverse of arccos(x) http://www.wolframalpha.com/input/?i=inverse+of+arccos%28x%29

Not only is the domain restricted to [0, pi] but where did those red imaginary logarithms/hyperbolas go???

Edited by MWresearch
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Is this what you wanted?

http://www.wolframalpha.com/input/?i=1%2F%28+arccos%28x%29%29

"Inverse of" would refer to swapping your range and domain around.

That's not at all what I was referring to, did you even read anything I said or just glance at the title?

I find it rude that I write all that and yet you can't give me the courtesy of reading it before posting.

LOOK at the wolfram alpha link. There's red lines. Those red lines are part of arccos(x). Those red lines can be represented more intuitively by the equation (-i*ln(i*x+sqrt(1-x^2)). Those red lines are some form of i*arccosh(i*x). Those red lines disappear when you take the inverse of arccos(x). Those red lines are not displayed when you plot the function cos(x). It is likely not a display error since obviously wolfram recognizes imaginary and complex values.

I never bothered to give anyone a -1 mark before but Jesus that's frustrating when I have to hold your hand to explain something you could easily understand yourself if you just read it.

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That's not at all what I was referring to, did you even read anything I said or just glance at the title?

I find it rude that I write all that and yet you can't give me the courtesy of reading it before posting.

LOOK at the wolfram alpha link. There's red lines. Those red lines are part of arccos(x). Those red lines can be represented more intuitively by the equation (-i*ln(i*x+sqrt(1-x^2)). Those red lines are some form of i*arccosh(i*x). Those red lines disappear when you take the inverse of arccos(x). Those red lines are not displayed when you plot the function cos(x). It is likely not a display error since obviously wolfram recognizes imaginary and complex values.

I never bothered to give anyone a -1 mark before but Jesus that's frustrating when I have to hold your hand to explain something you could easily understand yourself if you just read it.

I gave you it. For your poor attitude, which pervades in much of your questions. Nobody owes you anything and not everyone sees a question as the OP intended. Cest la vie.

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I gave you it. For your poor attitude, which pervades in much of your questions. Nobody owes you anything and not everyone sees a question as the OP intended. Cest la vie.

Obviously I don't give a crap what you think, I care about answers. If you have the audacity to call yourself any remote authority on any science or math, you better have the damn capacity to answer questions. Doesn't matter about anyone owing anything, it matters if this site can actually do what it set out to do.

Furthermore, the fact that you care about something as irrelevant as points is disturbing, this site needs to redefine who it calls "senior."

Here's how it works: people like me ask questions and legitimate scientists answer them because it makes them look good or feel validated. Put it on your resume, put it on a scholarship, I don't care, but that's the only reason for this site's existence.

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Whereas the notation f −1(x) might be misunderstood, f(x)−1 certainly denotes the multiplicative inverse of f(x) and has nothing to do with inversion of f.

https://en.wikipedia.org/wiki/Inverse_function

@OP: I read what you wrote, based on the fact that you didn't understand why "the domain restricted to [0, pi]" it led me to conclude you weren't aware that you may have asked Wolfram Alpha for something other than what you thought you did.

Honestly, you need to lose the attitude. People help those who are pleasant to deal with. This forum is for the discussion of Science, my dredging up memories of something I ran into over a decade ago is an optional extra.

If you still want the why the imaginary parts didn't appear then the answer is on the Wiki. I thought my response summed it up fairly succinctly but Wiki certainly has the answer if you feel the need to investigate further.

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https://en.wikipedia.org/wiki/Inverse_function

@OP: I read what you wrote, based on the fact that you didn't understand why "the domain restricted to [0, pi]" it led me to conclude you weren't aware that you may have asked Wolfram Alpha for something other than what you thought you did.

Honestly, you need to lose the attitude. People help those who are pleasant to deal with. This forum is for the discussion of Science, my dredging up memories of something I ran into over a decade ago is an optional extra.

If you still want the why the imaginary parts didn't appear then the answer is on the Wiki. I thought my response summed it up fairly succinctly but Wiki certainly has the answer if you feel the need to investigate further.

When in the hell did I state I didn't understand why the domain was restricted? Obviously you didn't read it. I merely pointed it out as something to consider, nowhere did I ask a question about the domain being restricted.

Not only that but the words "complex" and "imaginary" didn't occur a single time in your entire "solution to everything" article, but being pleasant obviously solved nothing because the only way to get any god damn attention around here is to shakes things up, otherwise my question would have been answered ages ago. No question answered? No reason to be pleasant. If it were different, that would be nice, but since the staff refused to answer the question I'm going to get someone else to or get enough attention until one of them does. Unfortunately for you, that person definitely isn't you. If there was actually a vertical line test failure in the domain [0,1] from taking the inverse of arccos(x) then obviously no function would be displayed at all in any way, it wouldn't force only the real component of the complex function to be displayed unless there was a complex function to begin with. But as it clearly shows with arccos(x) there are instances where the horizontal AND vertical line test creates complex values which in a normal circumstance should still exist when reflecting over the x axis. Not only that, but cos(z) in wolfram isn't even restricted by Re(z) when displaying cos(z), which means it's not an issue of software refusing to show those complex values, they simply disappear.

Edited by MWresearch
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Moderator Note

MWResearch - calm down and play nice.

Do not respond to this moderation

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Damn! I reached my negative vote limit for the day in this one thread in about 20 seconds. Never reached the limit before now. How abhorrent can a person be. lol. I think he must be doing it on purpose for some kind of humour that we are missing, no one can be that much of a dick.

MW - No one is going to help you with that attitude - go away and find somewhere else where you can be rude to people. I can think of many things I would like to say... but won't because I think it would break the forum rules.

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Where's the damn off button on you? How is it possible for your existence to be that big of a of time and resources? I don't give a crap what you think, if you don't have answers, no one cares, go ruin another website and let the legitimate grown up scientists do the discussing.

If you don't want to answer questions, don't waste everyone's time, your incessant trolling is exactly what loses credibility for this site.

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Moderator Note

On topic please. No more warnings.

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• 3 weeks later...

The answer, which, I'm surprised no one else came up with since there's supposedly people here who know a lot of math is that it doesn't disappear, there are complex and imaginary solutions of cos(x) at multiple points not visible in the real plane and it was simply not showing them since cos(x) generally only resides in the real plane.

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