Expert: Scotto - 3/4/2008

Question
Okay, this is something that has been bugging me for several months.  I would like to know what the exact formulas are for solving the reverse trig functions (ONLY reverse tan, reverse sin, and reverse cos please).  Every site I've found has talked about the identities, what they are in relation to another, how to use them on your calculator, how to find them by graphing a circle and drawing a bunch of lines.  PLEASE, forget all that stuff, I know about it, but I just want those three formulas.  
[Angle=(side a/side b)+*/- what ever other numbers]
Basically, I want y=b+c, that's all.  I've been to several other ask expert's sites, and all they've done is sent me the same crap I just wrote about.  Just shows they don't know squat.  I'm sorry that I seem so, "angered/passionate" about this, but I just want a straight answer.
   
                      Thank you very much,
                                Jacob

Answer
Take A as the angle of the hypoteneuse with far side a, near side b, and hypoteneuse c.  Note that C is a right angle and B = 90-A.

We have tan(A)=a/b, sin(A)=a/c, and cos(A)=b/c.
The arctan(a/b), the arcsin(a/c), and the arccos(b/c)
are all angle A.  

The only way to compute the arctan(x), arcsin(x), or arccos(x) is to input what they are equal to and compute the inverse on a calculator or look it up in a table.

To put them in polynomial format, you would need to use Newton's Method.  This is a summation from n=0 to infinity of f(0)f^n(0)/n!.
f^n(0) is the nth derivative and n! is n factorial, which is
n*(n-1)*(n-2)* ... 2*1.

There is no simple way to compute the arctan, arcsin, or arccos unless they are something simple like 0, root(2)/2, or 1.  They can be identified for some other values with a little computation, but not all values.

Once I had learned this, I began to see that mathematics was not as obvious as it was through algebra and geometry.  At least for algebra, the square root of a negative number can be computed once you get the idea of imaginary numbers down.