I'm working my way through basic trig (this question has a focus on inverse trig functions, specifically arcsine, arccosine and arctangent ), using Khan Academy, wikipedia and some of "trig without tears" - http://oakroadsystems.com/twt/
What I think I understand
My understanding is that the range of the usual principal value of arcsine, arccosine and arctangent is defined by "convention". Which reads to me, to mean a consensus of mathematicians agreed upon these principle values (I'm certain with some underlying reasoning, which I don't know of yet) to solve the issue of potential multiple results from a the same input to one of these functions.
What I don't get
Domain of arcsine & arccosine, why?
When discussing the domain of "x" for these 3 functions, as shown in this table on wikipedia: http://en.wikipedia.org/wiki/Inverse_trigonometric_function#Principal_values I see the domain of arcsine and arccosine is −1 ≤ x ≤ 1.
I've watched a number of basic tutorials on the unit circle and how it can be used to help solve these functions. So of course, I can see and visualize the fact that the maximum and minimum values of X on the unit circle are 1 & -1.
But I'm struggling to understand the intuition behind the restriction of the domain? My understanding right now is that the unit circle has x value [1;-1], so is that why the range is that?
Domain of arctangent, why?
Also the domain for arctangent is "all real numbers" - in the video on Khan Academy http://www.khanacademy.org/video/inverse-trig-functions--arctan?playlist=Trigonometry Sal (main teacher on Khan Academy) talks about how tangent of something also represents that slope of a line (I guess the hypotenuse) and how that could have infinite results.
I don't really understand this. If slope is rise over run - isn't there a limit to that ratio?