Chapter 3.7, Problem 19P

If multiply a complex number $z=r^{i\varphi }$ by $e^{i\theta }$, we get $e^{i\theta }z=r{e}^{i(\varphi +\theta )}$, that is, a complex number with the same r but with its angle increased by $\theta$.

We can say that the vector r from the origin to the point z = x + iy has been rotated by angle $\theta$ as in Figure 7.4 to become the vector R from the origin to the point Z = X + iY. Then we can write $X+iY=e^{i\theta }z=e^{i\theta }(x+iy)$.

Take real and imaginary parts of this equation to obtain equations (7.12).