Question

Write the polar form of the complex number $-3$

Answer 1

Hint: In this question, we need to write the complex number given by $x+iy$in terms of the polar form which can be written as $z=r\left( \cos \theta +i\sin \theta \right)$. Here, r is given by the formula $r=\sqrt{{{x}^{2}}+{{y}^{2}}}$and theta is given by $\theta =\arg \left( z \right)$. Now, the argument is given by the formula $\theta =\arg \left( z \right)={{\tan }^{-1}}\left( \dfrac{y}{x} \right)$. Here, as $x<0$ and $y>0$ so $\arg \left( z \right)=\pi -\theta$. Then on substituting the respective values and simplifying further we get the result.

Complete step-by-step answer:
Complex number:
A number of the form $z=x+iy$, where $x,y\in R$, is called a complex number.
The numbers x and y are respectively the real and imaginary parts of the complex number.
Polar Form:
If $z=x+iy$is a complex number, then z can be written as $z=r\left( \cos \theta +i\sin \theta \right)$, where $\theta =\arg \left( z \right)$ and $r=\sqrt{{{x}^{2}}+{{y}^{2}}}$this is called polar form.
Argument of a Complex Number:
Any complex number $z=x+iy$can be represented geometrically by a point (x , y) in a plane, called an Argand plane. The angle made by the line joining point z to the origin, with the X - axis is called the argument of that complex number. It is denoted by the symbol $\arg \left( z \right)$
$\theta =\arg \left( z \right)={{\tan }^{-1}}\left( \dfrac{y}{x} \right)$

Now, the given complex number in the question is $-3$
Now, on comparing this with the standard form of complex number $z=x+iy$we get,
$x=-3,y=0$
Now, as we already know that the formula for r is given by
$\Rightarrow r=\sqrt{{{x}^{2}}+{{y}^{2}}}$
Now, on substituting the respective values of x and y we get,
$\Rightarrow r=\sqrt{{{\left( -3 \right)}^{2}}+{{0}^{2}}}$
Now, on simplifying this further we get,
$\therefore r=3$
As we already know that the formula for argument is given by
$\Rightarrow \theta =\arg \left( z \right)={{\tan }^{-1}}\left( \dfrac{y}{x} \right)$
Here, we have a case that if $x<0$ and $y>0$, then
$\arg \left( z \right)=\pi -\theta$
Now, on substituting the respective values we get,
$\Rightarrow \theta =\pi -{{\tan }^{-1}}\left( \dfrac{0}{-3} \right)$
Now, this can be further written in the simplified form as
$\Rightarrow \theta =\pi -{{\tan }^{-1}}0$
Now, on further simplification we get,
$\therefore \theta =\pi$
Now, from the polar form representation of a complex number we have
$\Rightarrow z=r\left( \cos \theta +i\sin \theta \right)$
Now, on substituting the respective values we get,
$\therefore z=3\left( \cos \pi +i\sin \pi \right)$

Note:Instead of considering the argument of z as $\pi -\theta$if we consider the argument as only $\theta$then we get the corresponding value of the polar form incorrect. Because of simplifying the polar form we do not get the complex number that we considered.
It is important to note that while finding the polar form the argument of the complex number plays a vital role because the change in sign of the x and y values changes the theta value accordingly and so the polar form.