Question

Solve $\dfrac{1}{{1 + i}}$ in polar form?

Answer 1

Hint: In this question first, we will do the rationalization of the following complex number. Then we will convert the following complex number into the polar form by multiplying and dividing it by $\sqrt 2 $ so that we will convert it into a standard form.

Complete step by step answer:
In the above question, it is given that $\dfrac{1}{{1 + i}}$.
Now rationalize the above equation.
$ \Rightarrow \dfrac{1}{{1 + i}} \times \dfrac{{1 - i}}{{1 - i}}$
Here we will use the identity $\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$
$ \Rightarrow \dfrac{{1 - i}}{{{{\left( 1 \right)}^2} - {{\left( i \right)}^2}}}$
We know that ${i^2} = - 1$
$ \Rightarrow \dfrac{{1 - i}}{{1 - \left( { - 1} \right)}}$
$ \Rightarrow \dfrac{{1 - i}}{2}$
We can also write it as
$ \Rightarrow \dfrac{1}{2}\left( {1 - i} \right)$
Now multiply by $\sqrt 2 $ in the numerator and denominator.
$ \Rightarrow \dfrac{{\sqrt 2 }}{2}\dfrac{{\left( {1 - i} \right)}}{{\sqrt 2 }}$
$ \Rightarrow \dfrac{1}{{\sqrt 2 }}\left( {\dfrac{1}{{\sqrt 2 }} - i\dfrac{1}{{\sqrt 2 }}} \right)$
We know that $\sin \left( { - \dfrac{\pi }{4}} \right) = - \dfrac{1}{{\sqrt 2 }}$ and $\cos \left( { - \dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}$
Now substitute the above value
$ \Rightarrow \dfrac{1}{{\sqrt 2 }}\left( {\cos \left( { - \dfrac{\pi }{4}} \right) + i\sin \left( { - \dfrac{\pi }{4}} \right)} \right)$
Therefore, the polar form of $\dfrac{1}{{1 + i}}$ is $\dfrac{1}{{\sqrt 2 }}\left( {\cos \left( { - \dfrac{\pi }{4}} \right) + i\sin \left( { - \dfrac{\pi }{4}} \right)} \right)$.

Note:
The polar form of a complex number is a different way to represent a complex number apart from the rectangular form. Usually, we represent the complex numbers, in the form of z = x+ iy as in the above question where ‘i’ is the imaginary number. But in polar form, the complex numbers are represented as the combination of modulus and argument. The modulus of a complex number is also called absolute value or magnitude and the argument is the angle written inside the sine function. This polar form is represented with the help of polar coordinates of real and imaginary numbers in the coordinate system.