Complex Numbers

Polar Form of a Complex Number

The polar form or trigonometric form of a complex number is the representation of the complex number as the combination of its modulus and argument.

Let $z=(a,b)=a+ib$ be a complex number. It can be represented in the complex plane by a point $P$ with Cartesian coordinates $(a,b)$. Let $\theta$ be the angle in the standard position with $OP(=r)$ as its terminal arm.

Polar Form or Trigonometric Form of a Complex Number

Then, we have, \[\begin{array}{l} & \cos\theta & =\frac{a}{r} \\ \therefore & a &= r\cos\theta \\ \text{and,} & \sin\theta &= \frac{b}{r} \\ \therefore & b &= r\sin\theta \end{array}\]

Thus, the complex number $z=a+ib$ may be written in the following trigonometric form (polar form) \[\begin{array}{l} & z & =r\cos\theta+ir\sin\theta \\ \therefore & z &=r(\cos\theta+i\sin\theta)\end{array}\] \[\begin{array}{l} \text{where,} & r &= \sqrt{x^2+y^2} \\ \text{and,} & \tan\theta &= \frac{y}{x} & (x≠0) \end{array}\]

Here, $r=|z|$ is the modulus of $z$ and the angle $\theta$ is called the amplitude or the argument of $z$ and is written as $\text{amp}(z)$ or $\text{arg}(z)$.

Since $\sin\theta$ and $\cos\theta$ are both periodic with a period $2π$ or $360°$, the complex number \[z=a+ib=r(\cos\theta+i\sin\theta)\] may be written in the general form as \[z= r[\cos(\theta+2nπ)+i\sin(\theta+2nπ)\] \[\text{or,}\;\; z= r[\cos\theta(\theta+n360°+i\sin(\theta+n360°)]\] where, $n$ is an integer.

The polar (or trigonometric) form of the complex numbers is used in the computation of products and quotients of complex numbers. Other use includes the computation of powers and roots of complex numbers.