Question

What is the principal argument of $ \left( { - 1, - i} \right) $ , where $ i = \sqrt { - 1} $ ?
(A) $ \dfrac{\pi }{4} $
(B) $ - \dfrac{\pi }{4} $
(C) $ - \dfrac{{3\pi }}{4} $
(D) $ \dfrac{{3\pi }}{4} $

Answer 1

Hint:
Argument is also given by $ \theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right) $ , for a number $ \left( {x + iy} \right) $ . The principal argument is the unique value of the argument that lies in an interval $ \left( { - \pi ,\pi } \right) $ . Take the use of the inverse tangent function to find the correct answer from the given four options.

Complete step by step solution:
Here in this problem, we are given a complex number $ \left( { - 1, - i} \right) $ , where ‘iota’ is defined as $ i = \sqrt { - 1} $ . And with this information and using properties related to a complex number, we need to find the principal argument of this number.
Before starting with the solution we must understand a few things about complex numbers. A complex number is a number that is written as $ \left( {a + ib} \right) $ , in which “a” is a real number, and “b” is an imaginary number. The complex number contains a symbol “i” which satisfies the condition $ i = \sqrt { - 1} $ . Complex numbers can be referred to as the extension of the one-dimensional number line. In the complex plane, a complex number denoted by $ \left( {a + ib} \right) $ is represented in the form of the point $ \left( {a,b} \right) $ .
In polar form, a complex number is represented as $ r\left( {\cos \theta + i\sin \theta } \right) $ , where ‘r’ is known as modulus of the number and $ '\theta ' $ is the argument of the number. The argument is also given by $ \theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right) $ , for a number $ \left( {x + iy} \right) $ . According to this expression, an argument can obtain multiple values but a principal argument is only one and always lies in the interval $ \left( { - \pi ,\pi } \right) $ .
In this case, we have:
 $ \Rightarrow x + iy = - 1 - i \Rightarrow \theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right) = {\tan ^{ - 1}}\left( {\dfrac{{ - 1}}{{ - 1}}} \right) = {\tan ^{ - 1}}1 $
Now we know that
 $ \Rightarrow \tan \dfrac{\pi }{4} = 1 \Rightarrow \dfrac{\pi }{4} = {\tan ^{ - 1}}1 $
Thus we get the principal argument of the given complex number as $ \dfrac{\pi }{4} $

Hence, the option (A) is the correct answer.

Note:
An alternative approach can be taken to solve this problem by using the method of the Argand plane. Plot the given complex number on the Argand plane and find the angle made by the line joining the point to the origin. You can use trigonometric ratios to calculate the required argument.