The Inverse Sine  arcsin(z) .   Verify that the formula  

(i)               

is correct.  (At least for values of z in the upper half plane  .)

Explore Formula (ii).

First, use Mathematica to determine the formula for ArcSin[z].  Start with the identity  .  

And we can verify that    is the inverse.

We can use Mathematica to verify the formula graphically, for values of z in the upper half plane  .  

Remark.  If you mess around with the square root it will be wrong.  The portion that is supposed to be in the first quadrant appears symmetrically in the fourth quadrant.

(c) 2006 John H. Mathews, Russell W. Howell