🚀 go-pugleaf

On Wed, 23 Nov 1994, Andrew Kuemmel wrote:

> I thought that the number 2 does not have an inverse sine.
> But on the TI-85, when I press <2nd> <sin> 2 <enter>
> it gives an answer of (1.57...., -1.31...)

For complex numbers x+iy, the sine function is defined as
sin(x+iy) = sin(x)cosh(y)+icos(x)sinh(y).

                 y  -y                   y  -y
Note: cosh(y) = e +e      and sinh(y) = e -e
                ------                  ------
                  2                       2

When y=0, cosh(y)=1 and sinh(y)=0, so sin(x+i*0) = sin(x), the
function you are used to.  However, sin(Pi/2+1.316i)
= sin(Pi/2)cosh(1.316)+i*cos(Pi/2)sinh(1.316)
= cosh(1.316)+i*0 ~=~ 2.

Since the TI-85 "knows" all this, arcsin(2) gives Pi/2+1.316...i, which
the TI displays as (1.57... , 1.316...).

Final note:  all this is based on the identity

 i*y
e   = cos(y)+i*sin(y).


Mark Janeba			(503)370-6123
Dept. of Mathematics		e-mail:  mjaneba@willamette.edu
Willamette University		"Speaking for myself, not the university"
Salem, Oregon 97301-3922 USA