6. 3
The given complex number is − 3 .
Let r cos θ = − 3 (1)
and r sin θ = 0 (2)
Square and add equation (1) and equation (2).
( r cos θ ) 2 + ( r sin θ ) 2 = ( − 3 ) 2 + ( 0 ) 2 r 2 ( cos 2 θ + sin 2 θ ) = 9 + 0 r 2 = 9 r = ± 3
Since the modulus is always positive, therefore take positive value of r .
The value of modulus of complex variable is 3 .
Substitute 3 for r in equation (1).
3 cos θ = − 3 cos θ = − 1
Substitute 3 for r in equation (2).
3 sin θ = 0 sin θ = 0
As, cos θ is negative and sin θ is zero, therefore the value of θ is π .
The conversion of the complex number in polar form is,
z = r ( cos θ + i sin θ )
Substitute the values of z, r and θ in the above formula.
− 3 = 3 [ cos ( π ) + i sin ( π ) ]
Thus, the complex number − 3 in the polar form is 3 [ cos ( π ) + i sin ( π ) ] .
The given complex number is
Let
and
Square and add equation (1) and equation (2).
Since the modulus is always positive, therefore take positive value of
The value of modulus of complex variable is
Substitute 3 for r in equation (1).
Substitute 3 for r in equation (2).
As,
The conversion of the complex number in polar form is,
Substitute the values of z, r and
Thus, the complex number
