Question

Let ${{z}_{1}}$ and ${{z}_{2}}$ be two complex numbers with $\alpha ,\beta $ as their principal arguments such that $\alpha +\beta >\pi $, then principal $\arg \left( {{z}_{1}}{{z}_{2}} \right)$ is given by
A. $\alpha +\beta +\pi $
B. $\alpha +\beta -\pi $
C. $\alpha +\beta -2\pi $
D. $\alpha +\beta $

Answer 1

Hint: We explain the concept of argument for the complex number. Then we use the concept of range for argument and the theorem $\arg \left( {{z}_{1}}{{z}_{2}} \right)=\arg \left( {{z}_{1}} \right)+\arg \left( {{z}_{2}} \right)$ to find the $\arg \left( {{z}_{1}}{{z}_{2}} \right)$. We deduct $2\pi $ from the argument if it crosses $\pi $ to keep it in the range.

Complete step by step answer:
We have ${{z}_{1}}$ and ${{z}_{2}}$ as two complex numbers with $\alpha ,\beta $ as their principal arguments.
We know that $-\pi \le \alpha ,\beta \le \pi $. This range is for the argument of any complex number.
We can express any arbitrary complex number as $z={{e}^{i\theta }}$. Here $\theta $ is the argument.
We denote ${{z}_{1}}={{e}^{i\alpha }}$ and ${{z}_{2}}={{e}^{i\beta }}$. We also know that $\arg \left( {{z}_{1}}{{z}_{2}} \right)=\arg \left( {{z}_{1}} \right)+\arg \left( {{z}_{2}} \right)$.
Therefore, putting the values we get $\arg \left( {{z}_{1}}{{z}_{2}} \right)=\arg \left( {{z}_{1}} \right)+\arg \left( {{z}_{2}} \right)=\alpha +\beta $.
It is given that $\alpha +\beta >\pi $ which gives $\arg \left( {{z}_{1}}{{z}_{2}} \right)>\pi $.
But as the principal argument has to be in the range of $\left[ -\pi ,\pi \right]$, we deduct $2\pi $ from the argument if it crosses $\pi $ to keep it in the range. We can deduct as the period of the trigonometric function is $2\pi $.
Therefore, $\arg \left( {{z}_{1}}{{z}_{2}} \right)=\arg \left( {{z}_{1}} \right)+\arg \left( {{z}_{2}} \right)-2\pi $ if $\arg \left( {{z}_{1}} \right)+\arg \left( {{z}_{2}} \right)>\pi $.
So, principal $\arg \left( {{z}_{1}}{{z}_{2}} \right)$ is given by $\alpha +\beta -2\pi $.

So, the correct answer is “Option C”.

Note: The same thing can be done for condition of $\arg \left( {{z}_{1}} \right)+\arg \left( {{z}_{2}} \right)<-\pi $ by adding $2\pi $ to the argument if it goes less than $-\pi $ to keep it in the range. The complex form can also be represented as $z={{e}^{i\theta }}=\cos \theta +i\sin \theta $.