Question

Write the trigonometric form of $2\,-\,2\,i$ ?

Answer 1

In this question we have write trigonometric form of the given complex number form of any complex number first we have to find the values of modules of $z$ that is $\left| z \right|$ and Also, find the value of $\theta$. After getting these two values we will put these two values in polar form that is $r\left( \cos \theta +i\sin \theta \right)$. Where $r$ is nothing but value of $\left| z \right|$ after simplifying it we will get our required answer.

Complete step by step solution:
In this question we have given the complex number is $2\,-\,2\,i$
We will complex it with a complex number $z=a+ib$ and we get $a=2$ & $b=-2$ from a complex number.
In order to get trigonometric first we will find the value of $r$.
As we know that $r=\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}$ , where $z\,=\,a+ib$.

Which implies that $r=\sqrt{{{(2)}^{2}}+{{(-2)}^{2}}}$
Clearly, we can observe that $z=2-2i$ is represented by point $(+2,-2$ which lies on ${{4}^{th}}$ quadrant.
As we know that $\alpha ={{\tan }^{-1}}\left( \dfrac{Im(z)}{\operatorname{Re}(z)} \right)$
From $\left( z \right)$ here we have real part of $z$, that is $\operatorname{Re}(z)=2$ and imaginary part of $z$, that is $\operatorname{Im}(z)=-2$ which implies
Since we know that $\tan \left(\dfrac{\pi }{4} \right)=1$
Which implies $\alpha =-\dfrac{\pi }{4}$.
Now, rename $\alpha$ as $\theta$.
As we know that the equation of trigonometric function (polar form) is given as $r(\cos \theta +i\sin \theta )$ then
$\Rightarrow$The required trigonometric form of given complex number $2-2i$ is $2\sqrt{2}\left[ \cos\left(\dfrac{\pi }{4}\right) - i \sin\left(\dfrac{\pi }{4} \right) \right]$ .

Note: A complex number represented by an expression of the form $a+ib$ where $a$ and $b$ are real numbers.
Complex numbers can be represented as follows.
$z\,=\,a+ib$; where $a$ and $b$are real numbers.