Q.Express $e^{\cos\theta+i\sin\theta}$ in $a+ib$ form.

Rectangular form $z=x+iy$ is natural for addition/subtraction (just combine components), but multiplication, powers and roots are far easier in an alternate representation: polar form.

Polar coordinates. Superimposing polar coordinates $(r,\theta)$ — $r$ the distance from the pole $O$, $\theta$ the angle from the initial line, measured counter-clockwise — onto the rectangular Argand plane gives

so any nonzero $z=x+iy$ can be written

Here $r=|z|=\sqrt{x^2+y^2}$ is the modulus, and $\theta$ (found from $\tan\theta=y/x$, with the quadrant of $z$ fixing which angle) is an argument of $z$, written $\arg z$. Since adding any multiple of $2\pi$ to $\theta$ gives the same point, $\arg z$ has infinitely many values, all differing by $2k\pi$. The unique value with $-\pi<\theta\le\pi$ is the principal argument, $\operatorname{Arg}z$; every general argument is $\arg z=\operatorname{Arg}z+2k\pi,\ k\in\mathbb Z$. (For $z=0$, $\theta$ is undefined, so polar form always assumes $z\ne0$.) Conjugation flips the sign of the argument: if $z$ has polar coordinates $(r,\theta)$, $\overline z$ has $(r,-\theta)$.

Argument properties (mirroring the modulus properties): …