# Thread: help with this map

1. ## help with this map

Show that the mapping w = sin z maps the y-axis one-to-one and onto the v-axis.

2. Originally Posted by splash
Show that the mapping w = sin z maps the y-axis one-to-one and onto the v-axis.
I think you want to show it maps the y-axis one-to-one and onto the x-axis.

The y-axis are the imaginary numbers $\displaystyle xi$.

But that is not true,
$\displaystyle \sin (xi)=xi+\frac{x^3i}{3!}+\frac{x^5i}{5!}+\frac{x^7i }{7!}+...=i\sinh x$
It still remains an imaginary number. So you established no map.

3. Originally Posted by ThePerfectHacker
I think you want to show it maps the y-axis one-to-one and onto the x-axis.

The y-axis are the imaginary numbers $\displaystyle xi$.

But that is not true,
$\displaystyle \sin (xi)=xi+\frac{x^3i}{3!}+\frac{x^5i}{5!}+\frac{x^7i }{7!}+...=i\sinh x$
It still remains an imaginary number. So you established no map.
Where $\displaystyle w=u+\bold{i}v$, and $\displaystyle z=x+\bold{i}y$,
so now it does.

RonL