What's the visualisation of just $\sin\theta$ or $\cos\theta$ in complex plane.

Question

We know that if $z=\cos\theta+i\cdot \sin\theta\ $ then $z$ is a circle in complex plane. But what would $z=\cos\theta$ , $z=i\cdot \cos\theta$ , $z=\sin\theta$ or $z=i\cdot \sin\theta$ look like in complex plane?

Answer 1

We know that if $z=\cos\theta+i\cdot \sin\theta\ $ then $z$ is a circle in complex plane.

From this statement I infer that you're considering only real values for $\theta$.

In this case, it's pretty simple: the range of $\cos$ for real inputs is the line segment $[-1,1]$, the same goes for $\sin$, and if you multiply them by $i$ you obtain the line segment between $-i$ and $i$.

Answer 2

They are line segments. If $\theta$ has a range of at least $2\pi$, $z=\cos\theta$ and $z=\sin\theta$ are horizontal segments along the real axis, between $-1$ and $1$. Similarly $z=i\cos\theta$ and $z=i\sin\theta$ are segments along the imaginary axis, between $-i$ and $i$.

Answer 3

For any $\theta$ we have $-1\le \cos \theta \le 1$ (i.e. a segment on the real axis)

Similarly $z=i\cos \theta$ will be a segment on the immaginary axis, and the same idea for the other 2 curves.

But ofcourse $\sin$ and $\cos$ curves are on different positions for the same $\theta$ (for example $\cos 0=1$, but $\sin 0=0$