Rotate sine wave on complex plane

Question

Quick one. How do you rotate a sine wave on the complex plane? I already rotated the point $0+i$ to get the unit circle and graphed $n,e^{i \pi n}$ to get a sine wave, which is what all the examples are about. I now want to rotate the result at a $45^\circ$ angle. How do you do that?

Answer 1

You can rotate the complex plane 45 degrees counterclockwise (and any graph within it at the same time) by multiplying each point by $\frac{1+i}{\sqrt{2}}$.

You should be aware that, depending on your rotation, it may not be a graph of the real component of the complex number anymore, though.