write 1i and 1sqrt 3i in polar form and find their productSo

write 1+i and -1+sqrt (3i) in polar form and find their product.

Solution

1+i and -1 + sqrt 3i

polar form of 1+i

r= sqrt ( x^2 + y^2 ) = sqrt ( 1^2 + 1^2 ) = sqrt 2

theta = tan^-1 ( y/x) = tan^-1 ( 1) = 45 degrees

polar form = sqrt 2 ( cos 45 + i sin 45 )

-1 + sqrt 3i

r = sqrt ( 1^2 + 3 ) = 2

theta = tan^-1 ( sqrt 3 / 1) = 60 degrees

polar form = 2 ( cos 60 + i sin 60)

product = (-1-sqrt3) + i ( sqrt 3 -1)