The Inverse of a Complex Number in Polar Form

In this tutorial, let's explore the polar form of the multiplicative inverse of a Complex Number. We've established that the multiplicative inverse of a complex number is such that when we multiply a complex number, z by its inverse, we get the number 1, which is of course is a real number. So if we express z in its Cartesian form, i.e...

z = x + iy

then its inverse is given by...

z^{-1} = \frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2}

Now if we express z in polar form, we say z = rcisθ with r = |z|. And we can express this more formally as...

z = r\left ( \cos\theta +i\sin\theta \right )=r\cos\theta+ir\sin\theta

Now, we note that x = rcosθ and y = rsinθ, so if we substitute these expressions for x and y, into the inverse of z, we should get...

z =\frac{r\cos\theta}{r^2\cos^2\theta+r^2\sin^2\theta}-i\frac{r\sin\theta}{r^2\cos^2\theta+r^2\sin^2\theta}

Please watch the video for the full tutorial!