Calculator test

[Albert Chan]

(Yesterday 01:56 PM)Commie Wrote:  Here is one of the equations for tan(x+iy):
\[\tan(x+iy) = \frac{sin{2x} + {i}sinh{2y}}{cosh{2y}+cos{2x}}\]

Proof is trivial, using sum to product formula. Let Y = i*y

RHS
= (sin(2x) + sin(2Y)) / (cos(2x) + cos(2Y))
= (2*sin(x+Y)*cos(x-Y)) / (2*cos(x+Y)*cos(x-Y))
= sin(x+Y) / cos(x+Y)
= tan(x+i*y)

Note that cos(2x) has range [-1,1], cosh(2y) has range [1,inf)
RHS denominator may cause catastrophic cancellation.

This was the reason Free42 tan(z) later switched to this:

tan(x+i*y) = (sin(x)*cos(x) + i * cosh(y)*sinh(y)) / (cos(x)^2 + sinh(y)^2)