Question

The mod – amp form of $-1-\sqrt{3}i$ is:
A. $cis\left( \dfrac{2\pi }{3} \right)$
B. $2cis\left( -\dfrac{2\pi }{3} \right)$
C. $cis\left( -\dfrac{2\pi }{3} \right)$
D. $2cis\left( \dfrac{2\pi }{3} \right)$

Answer 1

A complex number is a number that can be expressed in the form $a+ib$, where a, b are real numbers and i is an imaginary number equal to $\sqrt{-1}$. The mod – amp form of this number is given as $Z=r.cis\left( \theta \right)$, where $r=\left| \sqrt{{{a}^{2}}+{{b}^{2}}} \right|$ and $\theta ={{\tan }^{-1}}\left( \dfrac{b}{a} \right)$.

Complete step by step answer:
Let us first understand what is a complex number. A complex number is a number that can be expressed in the form $a+ib$, where a, b are real numbers and i is an imaginary number called iota and it is equal to $\sqrt{-1}$.The complex numbers are denoted by the letter Z.Every complex number can be expressed in another form called mod – amp form. Suppose we have a complex number $Z=a+ib$. Then the mod – amp form of this number is given as $Z=r.cis\left( \theta \right)$, where r is called the modulus of the complex number and the angle $\theta$ is the angle the complex number makes with the real axis in the real – imaginary axes plane and it is called the argument of the complex number.

Here, $r=\left| \sqrt{{{a}^{2}}+{{b}^{2}}} \right|$
And $\theta ={{\tan }^{-1}}\left( \dfrac{b}{a} \right)$, where $0\le \theta \le \pi$ or $0\ge \theta \ge -\pi$.
The complex number given in the question is $-1-\sqrt{3}i$.
This means that $a=-1$ and $b=-\sqrt{3}$.
Therefore, the modulus of the complex number is,
$r=\left| \sqrt{{{a}^{2}}+{{b}^{2}}} \right|\\\ \Rightarrow r=\left| \sqrt{{{(-1)}^{2}}+{{\left( -\sqrt{3} \right)}^{2}}} \right|$
$\Rightarrow r=\left| \sqrt{1+3} \right|=\left| \sqrt{4} \right|=2$

Now, the argument of the complex number is,
$\theta ={{\tan }^{-1}}\left( \dfrac{-\sqrt{3}}{-1} \right)$
$\Rightarrow \theta ={{\tan }^{-1}}\left( \sqrt{3} \right)$
Since a and b are both negative, this means that complex numbers lie in the third quadrant.Therefore, we get that
$\theta ={{\tan }^{-1}}\left( \sqrt{3} \right)\\\ \Rightarrow\theta=-\pi +\dfrac{\pi }{3}\\\ \therefore\theta=-\dfrac{2\pi }{3}$
Therefore, the mod – amp form of the complex number $2cis\left( -\dfrac{2\pi }{3} \right)$.

Hence, the correct option is B.

Note: Students must be careful while calculating the values of the modulus (r) and argument ($\theta$).Note that the value of r is always positive. While calculating the value of $\theta$, students must keep in mind about the rage of values that the argument of a complex number can take. The values must strictly lie within the range.