Question

If $\arg z<0$, then $\arg \left( -z \right)-\arg z=$
(A) $\pi$
(B) $-\pi$
(C) $\dfrac{\pi }{2}$
(D) $-\dfrac{\pi }{2}$

Answer 1

We are given an expression $\arg z<0$ and using this we have to compute the value for $\arg \left( -z \right)-\arg z$ from the given options. We will first write the let that ‘z’ in the given expression be, $z=r\left( \cos \theta +i\sin \theta \right)$. So, we will have $\arg \left( z \right)=\theta$. We will use this form to find the value of $\arg \left( -z \right)$. We know that, $\cos \left( \pi +\theta \right)=-\cos \theta$ and $\sin \left( \pi +\theta \right)=-\sin \theta$. Then, we will compute the values in $\arg \left( -z \right)-\arg z$ and get the required value.

Complete step by step answer:
According to the given question, we are given an expression $\arg z<0$ and we are asked to use this expression and find the value of $\arg \left( -z \right)-\arg z$.
We have the given expression as
$\arg z<0$
Let us assume the variable in the given expression be written in terms of polar coordinate form and we get,
$z=r\left( \cos \theta +i\sin \theta \right)$
And $\arg \left( z \right)=\theta <0$
We will now find the value of $\arg \left( -z \right)$, and for that we have,
$-z=-r\left( \cos \theta +i\sin \theta \right)$
In the above equation, the negative sign in the RHS has the cosine function and the sine function as negative. But we know that, $\cos \left( \pi +\theta \right)=-\cos \theta$ and $\sin \left( \pi +\theta \right)=-\sin \theta$. So, we can write the expression as,
$\Rightarrow -z=r\left( \cos \left( \pi +\theta \right)+i\sin \left( \pi +\theta \right) \right)$
So, if we write $\arg \left( z \right)=\theta$ for the expression $z=r\left( \cos \theta +i\sin \theta \right)$, that is, we write the function in terms of the angle and so for the expression $-z=r\left( \cos \left( \pi +\theta \right)+i\sin \left( \pi +\theta \right) \right)$, we will have the function as,
$\arg \left( -z \right)=\pi +\theta$
Now, we will substitute the obtained values in the expression,
$\arg \left( -z \right)-\arg z$
We get,
$\Rightarrow \left( \pi +\theta \right)-\theta$
Opening up the brackets, we get the value of the expression as,
$\Rightarrow \pi +\theta -\theta$
$\Rightarrow \pi$

So, the correct answer is “Option A”.

Note: The given function $\arg \left( z \right)$is given to have a value less than 0, that is, existence of imaginary terms/ numbers. That is why we introduced the term iota ($i$) while defining the value of the variable ‘z’. So, if the value of the function is greater than 0 then we will have real values so the iota ($i$) is not required.