Verified answer

1)
[tex]\boxed{\begin{matrix} \text{Seja o complexo }: \\\\ \sf z = a+b\cdot i \\\\ \text{m\'odulo do complexo z} : \\\\ \sf |z| = \sqrt{a^2+b^2}\\\\ \text{Forma polar/trigonom\'etrica do complexo z} : \\\\ \displaystyle \sf z=|z|\cdot \underbrace{\sf \left[ cos(\theta)+i\cdot sen(\theta) \right] }_{\sf cis(\theta)}\\\\ \sf z = |z|\cdot cis(\theta) \end{matrix}}[/tex]

(2)
[tex]\boxed{\begin{matrix}\text{Propriedade de complexos }: \\\\ \sf cis(\theta)\cdot cis(\alpha) = cis(\theta+\alpha) \\\\ \displaystyle \sf cis(\theta)^{n}=\underbrace{\sf cis(\theta)\cdot cis(\theta)\cdot cis(\theta) + ... }_{\text{n vezes}} = cis(n\cdot \theta) \end{matrix}}[/tex]


comentário :

[tex]\displaystyle \sf cis(\theta) = cos(\theta)+i\cdot sen(\theta) \\\\ \text{Fa\c camos}: \\\\\ C =cos(a_o)+cos(a_o+r)+cos(a_o+2r)+...+cos(a_o+(n-1)r) \\\\ i\cdot S = i\cdot sen(a_o)+i\cdot sen(a_o+r)+i\cdot sen(a_o+2r)+..+i\cdot sen(a_o+(n-1)r) \\\\\ C+i\cdot S = cos(a_o)+i\cdot sen(a_o)+...+cos(a_o+(n-1)r +i\cdot sen(a_o+(n-1)r) \\\\\ C+i\cdot S = cis(a_o)+cis(a_o+r) +cis(a_o+2r)+...+cis(a_o+(n-1)r)[/tex]

[tex]\displaystyle \sf \text{note que} : \\\\ cis(a_o+r) = cis(a_o)\cdot cis(r) \\\\ cis(a_o+2r) = cis(a_o)\cdot cis(2r)\\ ....\\\ cis(a_o+(n-1)r)=cis(a_o)\cdot cis((n-1)r)[/tex]  

Note que se trata de uma P.G de razão cis(r)
então :

[tex]\displaystyle \sf C+i\cdot S = \underbrace{\sf cis(a_o)+cis(a_o+r) +cis(a_o+2r)+...+cis(a_o+(n-1)r)}_{\displaystyle \text{Soma de uma P.G de raz\~ao cis(r) }} \\\\\\ C+i\cdot S = \frac{cis(a_o)\cdot [(cis(r))^{n} -1]}{cis(r)-1} \\\\\\ C +i\cdot S = \frac{cis(a_o)\cdot [cis(n\cdot r)-1]}{cis(r)-1}[/tex]

[tex]\displaystyle \sf C+i\cdot S = \frac{\displaystyle \sf cis(a_o)\cdot cis\left( \frac{n\cdot r}{2}\right)\left[ cis\left(\frac{n\cdot r }{2}\right)-cis\left(\frac{-n\cdot r }{2}\right) \right]}{\displaystyle \sf cis\left(\frac{r}{2}\right)\cdot \left[cis\left(\frac{r}{2}\right)-cis\left(\frac{-r}{2}\right) \right]}[/tex]

[tex]\displaystyle \sf obs: \\\\ cis\left(\frac{n\cdot r }{2}\right)-cis\left(\frac{-n\cdot r }{2}\right) =cis\left(\frac{ r }{2}\right)^{n}-cis\left(\frac{\cdot r }{2}\right) ^{-n} = 2\cdot i\cdot sen\left(\frac{n\cdot r}{2}\right) \\\\\\\ cis\left(\frac{ r }{2}\right)-cis\left(\frac{- r }{2}\right) = cis\left(\frac{ r }{2}\right)^{1}-cis\left(\frac{ r }{2}\right)^{-1} = 2\cdot i\cdot sen\left(\frac{r}{2}\right)[/tex]

Daí :

[tex]\displaystyle \sf C+i\cdot S = \frac{cis(a_o)\cdot \displaystyle \sf cis\left(\frac{n\cdot r}{2}\right)\cdot 2\cdot i\cdot sen\left(\frac{n\cdot r}{2}\right) }{\displaystyle \sf cis\left(\frac{r}{2}\right)\cdot 2\cdot i\cdot sen\left(\frac{r}{2}\right) } \\\\\\ C+i\cdot S = \frac{\displaystyle cis(a_o) \cdot cis\left(\frac{n\cdot r }{2}\right)\cdot sen\left(\frac{n\cdot r}{2}\right)}{\displaystyle cis\left(\frac{r}{2}\right)\cdot sen\left(\frac{r}{2}\right) }[/tex]

[tex]\displaystyle \sf C+i\cdot S = \frac{\displaystyle cis\left(a_o+\frac{n\cdot r }{2}-\frac{r}{2} \right)\cdot sen\left(\frac{n\cdot r}{2}\right)}{\displaystyle sen\left(\frac{r}{2}\right) } \\\\\\ C+i\cdot S =\frac{\displaystyle cis\left(a_o+(n-1)\cdot \frac{1}{2}\right)\cdot sen\left(\frac{n\cdot r }{2}\right) }{\displaystyle sen\left(\frac{r}{2}\right) }[/tex]

[tex]\displaystyle \sf C+i\cdot S =\frac{\displaystyle \left[cos\left(a_o+(n-1)\cdot \frac{1}{2}\right)+i\cdot sen\left(a_o+(n-1)\cdot \frac{1}{2}\right)\right] \cdot sen\left(\frac{n\cdot r }{2}\right) }{\displaystyle sen\left(\frac{r}{2}\right) }[/tex]
C é a parte real e S é a parte imaginária.
Queremos a sequência do seno, então vamos igualar imaginária com parte imaginária

[tex]\displaystyle \sf S = \frac{\displaystyle sen\left(a_o+(n-1)\cdot \frac{r}{2}\right)\cdot sen\left(\frac{n\cdot r}{2}\right)}{\displaystyle sen\left(\frac{r}{2}\right)} \\\\\\ S = cossec\left(\frac{r}{2}\right)\cdot sen\left(a_o+(n-1)\cdot \frac{r}{2}\right)\cdot sen\left(\frac{n\cdot r}{2}\right)[/tex]

mas

[tex]\displaystyle \sf S = sen(a_o)+sen(a_o+r)+...+sen(a_o+(n-1).r) \\\\ S = \sum^{n}_{k = 0 }sen\left(a_o+k\cdot r\right)[/tex]

[tex]\boxed{\ \displaystyle \sf \sum^{n}_{k = 0 }sen\left(a_o+k\cdot r\right) =cossec\left(\frac{r}{2}\right)\cdot sen\left(a_o+(n-1)\cdot \frac{r}{2}\right)\cdot sen\left(\frac{n\cdot r}{2}\right) \ }[/tex]