Images of lines $y = k = \mbox{constant}$ under the mapping $w = \cos (z)$

Question

I want to solve this question: find the images of lines $y = k = \mbox{constant}$ under the mapping $w =\cos(z).$ I know that $w=\cos(z)=\cos(x)\cosh(y)-i\sin(x)\sinh(y)$ so $u=\cos(x)\cosh(y)$ and $v=-\sin(x)\sinh(y)$ but $y=k$ is only a line and how can i map it? I don't know what values should be given to x. I put k in $u=\cos(x) \cosh(k)$ and in $v=-\sin(x)\sinh(k).$ Lastly, I ask what should I do?

Answer 1

As Daniel Fisher suggested you should think of $\cosh(k)$ and $\sinh(k)$ as constants. For example $a$ en $b$.

The resulting function $x\mapsto a\cos x-i b\sin x$ should ring a bell.

Perhaps it's more clear if explicitly written as a coordinate.

$$w(t) = (a\cos t, -b\sin t) \qquad t\in [0,2\pi]$$

(since the function is $2\pi$ periodic)

If $a=b$ then you'd find a circle. When they are not equal you find an ellips. Perhaps you remember the cartians form. Just set $x=a\cos t$ and $y=-b\sin t$, then

$$\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$$

Here's a picture of the map. Bigger values of $k$ make the ellipse more circle-like. (since $\cosh k$ would be closer to $\sinh k$).