Complex analysis for R6RS implementation

Definitions

$e^{i\theta}=cos\theta+isin\theta$
$z=x+iy=re^{i\theta}$
$r=\sqrt{x^2+y^2}$
$\theta=arctan(\frac{y}{x})$
$cos(iy) = \frac{e^{-y} + e^y}{2}$
$sin(iy) = \frac{e^{y} - e^{-y}}{2i}$

sin(z)

$sin(z)$
$=sin(x+iy)$
$= sin(x)cos(iy) + sin(iy)cos(x)$
$= sin(x)\frac{e^{-y} + e^y}{2} + cos(x)\frac{e^{y} - e^{-y}}{2i}$
$= sin(x)\frac{e^{-y} + e^y}{2} - cos(x)\frac{e^y - e^{-y}}{2}i$

cos(z)

$cos(z)$
$=cos(x+iy)$
$=cos(x)cos(iy) - sin(x)sin(iy)$
$=cos(x)\frac{e^{-y} + e^y}{2} - sin(x)\frac{e^{y} - e^{-y}}{2i}$
$=cos(x)\frac{e^{-y} + e^y}{2} + sin(x)\frac{e^{y} - e^{-y}}{2}i$

tan(z)

$tan(z)=\frac{sin(z)}{cos(z)}$

arcsin(z)

$arcsin(z)=\phi$
$\Rightarrow sin(arcsin(z))=sin(\phi)$
$\Rightarrow z=sin(\phi)$
$\Rightarrow z=\frac{e^{i\phi} - e^{-i\phi}}{2i}$
$let X=e^{i\phi}$
$\Rightarrow z=\frac{X-\frac{1}{X}}{2i}$
$\Rightarrow X^2-2izX - 1=0$
$\Rightarrow X=iz \pm \sqrt{1-z^2}$
$\Rightarrow e^{i\phi} = iz \pm \sqrt{1-z^2}$
$\Rightarrow log{e^{i\phi}} = log(iz \pm \sqrt{1-z^2})$
$\Rightarrow i\phi = log(iz \pm \sqrt{1-z^2})$
$\Rightarrow arcsin(z) = \phi = -ilog(iz \pm \sqrt{1-z^2})$