Question

The $\arg \left( { - \dfrac{3}{2}} \right)$ equals
A. $\dfrac{\pi }{2}$
B. $- \dfrac{\pi }{2}$
C.0
D. $\pi$

Answer 1

Hint : Here in this question, we have to find the angle of the complex number using a given argument number. As we know the complex number is defined as $z = x + iy$ , where $x = r\cos \theta$ , $y = r\sin \theta$ and $i$ be the imaginary number by giving the value of r to the polar form of complex number $z = r\left( {\cos \theta + i\sin \theta } \right)$ using a given argument number we get the angle $\theta$ .

Complete step-by-step answer :
The argument of a complex number is defined as the angle inclined from the real axis in the direction of the complex number represented on the complex plane. It is denoted by “ $\theta$ ”. It is measured in the standard unit called “radians”.
In polar form, a complex number is represented by the equation $z = r\left( {\cos \theta + i\sin \theta } \right)$ , here, $\theta$ is the argument. The argument function is denoted by $\arg \left( z \right)$ , where z denotes the complex number, i.e., $z = x + iy$ . The computation of the complex argument can be done by using the following formula:
i.e., $\arg \left( z \right) = \theta$
Therefore, the argument θ is represented as: $\theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)$
Now, consider the given question
$\Rightarrow \,\,\arg \left( z \right) = \arg \left( { - \dfrac{3}{2}} \right)$
by
Where, z is the complex number i.e., $z = x + iy$ , then
$\Rightarrow \,\,\arg \left( {x + iy} \right) = \arg \left( { - \dfrac{3}{2}} \right)$
Let us take
$\Rightarrow \,\,x + iy = \left( { - \dfrac{3}{2}} \right)$
Put, $x = r\cos \theta$ and $y = r\sin \theta$ , then on substituting we have
$\Rightarrow \,\,r\cos \theta + i\,r\sin \theta = \left( { - \dfrac{3}{2}} \right)$
Take r as common in LHS, then
$\Rightarrow \,\,r\left( {\cos \theta + i\,\sin \theta } \right) = \left( { - \dfrac{3}{2}} \right)$
Now, put $r = \dfrac{3}{2}$ and $\theta = \pi$ , then
$\Rightarrow \,\,\dfrac{3}{2}\left( {\cos \left( \pi \right) + i\,\sin \left( \pi \right)} \right) = \left( { - \dfrac{3}{2}} \right)$
By the standard trigonometric table the value of $\cos \left( \pi \right) = - 1$ and $\sin \left( \pi \right) = 0$ , on substituting the values we have
$\Rightarrow \,\,\dfrac{3}{2}\left( { - 1 + i\,\left( 0 \right)} \right) = \left( { - \dfrac{3}{2}} \right)$
$\Rightarrow \,\,\dfrac{3}{2}\left( { - 1} \right) = \left( { - \dfrac{3}{2}} \right)$
$\Rightarrow \,\, - \dfrac{3}{2} = - \dfrac{3}{2}$
Hence, $\arg \left( { - \dfrac{3}{2}} \right) = \pi$
Therefore, option (D) is correct.
So, the correct answer is “Option D”.

Note : A complex number are one of the numbers that are expressed in the form of $a + ib$ , where a,b be the real number and $i$ be an imaginary number, absolute number is an angle towards the direction of the complex number it can easily find by a formula of $\theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)$ , where, $y = r\sin \theta$ and $x = r\sin \theta$ .