Q.Convert $r(\cos\theta+i\sin\theta)$ back to the form $a+ib$ for $r=2,\ \theta=\dfrac{\pi}{3}$.

Substituting $a=r\cos\theta$ and $b=r\sin\theta$ (where $r=|z|$ and $\theta=\arg(z)$) into $z=a+ib$ gives the polar form $z=r(\cos\theta+i\sin\theta)$. The ordered pair $(r,\theta)$ is called the polar coordinates of the point representing $z$, describing the very same point that the Cartesian pair $(a,b)$ describes, just measured as a distance-and-direction from the origin instead of as horizontal-and-vertical offsets. Converting from Cartesian to polar form means computing the modulus and the quadrant-corrected argument as usual and substituting them in; converting from polar back to Cartesian mea …