# Thread: polar forms

1. ## polar forms

how can i express (1+i)/(sqrt(3)+i) in the x+iy form? Also, using those two smaller expressions, i must show that cos(pi/12) = (sqrt(3)+1)/(2sqrt(2)) and sin(pi/12) = (sqrt(3)-1)/(2sqrt(2))

another one, i must show that if w is an nth root of unity, then w = 1/w. deduce that (1-w)^n = (w-1)^n. (showing that (1-w)^2n is real)
the underscores go above the numbers, to show the conjugate

2. The quotient of two complex numbers can be obtained by dividing the moduli and subtrating the arguments.

3. For any complex number we have $\frac{1}{z} = \frac{{\overline z }}{{\left| z \right|^2 }}$.
So we have: $\frac{{1 + i}}{{\sqrt 3 + i}} = \left( {1 + i} \right)\frac{{\sqrt 3 - i}}{2} = \left( {\frac{{\sqrt 3 + 1}}{2}} \right) + i\left( {\frac{{\sqrt 3 - 1}}{2}} \right)$.

Now if w is an nth root of unity, from the above we see that $\frac{1}{w} = \frac{{\overline w }}{{\left| w \right|^2 }} = \overline w$.
In general you need to show: $\overline {\left( z \right)} ^n = \left( {\overline z } \right)^n$.
Put together we get:
$\overline {\left( {1 - w} \right)} ^n = \left( {1 - \overline w } \right)^n = \left( {1 - \frac{1}{w}} \right)^n = \left( {\frac{{w - 1}}{w}} \right)^n = \frac{{\left( {w - 1} \right)^n }}{{w^n }} = \left( {w - 1} \right)^n$