Write in Polar Form i(i+1)(-√3+i)

Question

I managed to write i; (i+1); and (-√3+i) in polar form. I got r=√2 r=2 and r =√1, respectively. I then solved for θ and got

π/4 = 3π/12,

5π/6 --> 6π/6 - 5π/6 = π/6 = 2π/12

and π/2=6π/12 respectively.

and I got (cos(11π/12) + i sin(11π/12) for my answer.

but the answer in my book is: (cos(19π/12) + i sin(19π/12)

the part I'm stuck on is why my book left 5π/6 in the second quadrant, shouldn't they have subtracted π? any help would be appreciated.

Answer 1

Draw and label a diagram for each!

i, (1, π/2)

1 + i, (√2, π/4)

-√3 + i, (2, 5π/6)