The Inverse Cosine  arccos(z) .   Verify that the formula(s)   

(ii a)            ,  

(ii a)            .  

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

Explore Formula (ii a).

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

The above formula looks different from ours.  However, the following simplification can be made.

  

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. A portion that is supposed to be in the fourth quadrant appears symmetrically in the second quadrant.

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