6. Rectangular and Polar Forms

L3CALC

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Every complex number that we’ve used, in the form of $z=x+iy$ is expressed in the ‘rectangular form’.

There is one more form in Complex Numbers — known as the ‘polar form’. This consists of using $\text{cis}$, and the angle between the positive real axis, and the modulus line.

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For the image above, we can see that $r$ is the modulus, and $\theta$ is the angle between the positive real axis, and the modulus line.

The point $z$ $(r,\theta )$‘s argument is $\theta$, and can be expressed as $\text{arg}(z)=\theta$.

Using triginometry, we know that the side $a$ (adjacent), and $b$ (opposite) are respectively $r\cos \theta$, and $r\sin \theta$.

However, using the Argand diagram we know that the vertical is imaginary, and therefore $b$ would be $r\times i\sin \theta$.