Question

1+ 1

Answer 1

The given complex number is −1+i.

Let rcosθ=−1(1)

and rsinθ=1(2)

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

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

Since the 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 cosθ=−1 cosθ=− 1 2 θ=− π 4

Substitute 2 for r in equation (2).

2 sinθ=1 sinθ= 1 2 θ= π 4

As cosθ is negative and sinθ is positive, therefore 'θ' lies in the second quadrant.

So, the value of θ is ( π− π 4 )= 3π 4 .

The conversion of the complex number in polar form is,

z=r( cosθ+isinθ )

Substitute the values of z, r and θ in the above formula.

−1+i= 2 [ cos( 3π 4 )+isin( 3π 4 ) ]

Thus, the complex number −1+i in the polar form is 2 [ cos( 3π 4 )+isin( 3π 4 ) ].