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

Hint: 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.