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 \[x = r\sin \theta \] .