Question

Find the modulus and the argument of the complex number

Answer 1

The given complex number is z=−1− 3 i.

Let rcosθ=−1(1)

and rsinθ=− 3 (2)

Square and add equation (1) and equation (2).

( rcosθ ) 2 + ( rsinθ ) 2 = ( −1 ) 2 + ( − 3 ) 2 r 2 ( cos 2 θ+ sin 2 θ )=1+3 r 2 =4 r=±2

Since, modulus is always positive, therefore take positive value of r.

The value of modulus of the complex variable is 2.

Substitute 2 for r in equation (1).

2cosθ=−1 cosθ= −1 2 θ=− π 3

Substitute 2 for r in equation (2).

2sinθ=− 3 sinθ= − 3 2 θ=− π 3

Since, both the values of sinθ and cosθ are negative and lie in the third quadrant. Therefore,

Argument=−( π− π 3 ) = −2π 3

Thus, the modulus of the complex number −1−3i is 2 and argument is −2π 3 .