Question

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

Answer 1

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 $\arg z = {\tan ^{ - 1}}\dfrac{y}{x}$ .