Question

If \[\arg z = \theta \] , then \[\arg \overline z = \]
A) \[\theta - \pi \]
B) \[\pi - \theta \]
C) \[\theta \]
D) \[ - \theta \]

Answer 1

Hint: Any complex number \[z\] can be written as \[z = \cos \theta + i\sin \theta \] ,where \[\theta \] is called the argument of \[z\] which is defined as the angle between the positive real axis and the line joining the origin and the point. Also, \[\overline z \] is called the conjugate of \[z\] . The conjugate of \[z\] has the same real part but the imaginary part with the opposite sign .i.e. if \[z = x + iy\] then \[\overline z = x - iy\] . Now, in this question, we need to find the value of the argument of \[\overline z \] . So, we will first write \[z = \cos \theta + i\sin \theta \] and then find its conjugate. After finding the conjugate of \[z\] , we will write \[\overline z \] as \[\overline z = \cos \phi + i\sin \phi \] , where \[\phi \] is the argument of \[\overline z \] .

Complete answer:
We are given \[\arg z = \theta \]
\[\therefore \] , \[z\] can be written as \[z = \cos \theta + i\sin \theta \]
Let us suppose \[\arg \overline z = \phi \]
We now find the conjugate of \[z\] i.e. \[\overline z \]
Now, since \[z = \cos \theta + i\sin \theta \]
\[\overline z = \cos \theta - i\sin \theta \] ------(1)
Now, we have to write \[\overline z \] as \[\overline z = \cos \phi + i\sin \phi \]
From (1)
 \[\therefore \overline z = \cos \theta + i( - \sin \theta )\] -------(2)
We know,
\[\cos ( - \theta ) = \cos \theta \]
\[\sin ( - \theta ) = - \sin \theta \]
So, (2) can be written as
\[\overline z = \cos ( - \theta ) + i\sin ( - \theta )\]
Now, comparing \[\overline z = \cos ( - \theta ) + i\sin ( - \theta )\] and \[\overline z = \cos \phi + i\sin \phi \] , we get
\[\phi = - \theta \]
\[\therefore \arg \overline z = \phi = - \theta \]
Hence, we get ,
If \[\arg z = \theta \] , then \[\arg \overline z = - \theta \]
Therefore, option (D) is the correct answer.

Note:
We need to take care of the fact that \[z\] is always written as \[z = \cos \theta + i\sin \theta \] ,i.e. there is always a ‘+’ sign in between and not ‘-‘ sign. Also, we should know that \[\cos ( - \theta ) = \cos \theta \] and \[\sin ( - \theta ) = - \sin \theta \] . While comparing the given condition and the obtained condition we need to take care of the sign. We can also use the formula for finding the argument of \[z\] i.e. if \[z = x + iy\] , then \[\arg z = {\tan ^{ - 1}}\dfrac{y}{x}\] .