Find the modulus and the argument of the complex number
The given complex number is z = − 3 + i .
Let r cos θ = − 3 (1)
and r sin θ = 1 (2)
Square and add equation (1) and equation (2).
( r cos θ ) 2 + ( r sin θ ) 2 = ( − 3 ) 2 + ( 1 ) 2 r 2 ( cos 2 θ + sin 2 θ ) = 3 + 1 r 2 = 4 r = ± 2
Since, modulus is always positive, therefore take positive value of r .
The value of the modulus of the complex variable is 2 .
Substitute 2 for r in equation (1).
2 cos θ = − 3 cos θ = − 3 2 θ = − π 6
Substitute 2 for r in equation (2).
2 sin θ = 1 sin θ = 1 2 θ = π 6
Since, the value of sin θ is positive and cos θ is negative and θ lies in second quadrant. So,
Argument = ( π − π 6 ) = 5 π 6
Thus, the modulus of the complex number − 3 + i is 2 and an argument is 5 π 6 .
The given complex number is
Let
and
Square and add equation (1) and equation (2).
Since, modulus is always positive, therefore take positive value of
The value of the modulus of the complex variable is
Substitute 2 for r in equation (1).
Substitute 2 for r in equation (2).
Since, the value of
Thus, the modulus of the complex number
