Tendo dois complexos quaisquer :
[tex]\text z_1 = |\text z_1|.[\ \text{Cos}(\theta) + \text{i.Sen}(\theta) \ ][/tex]
[tex]\text z_2 = |\text z_2|.[\ \text{Cos}(\alpha) + \text{i.Sen}(\alpha) \ ][/tex]
Temos que o produto vale :
[tex]\text z_1.\text z_2 = |\text z_1|.|\text z_2|.[\ \text{Cos}(\theta+\alpha) + \text{i.Sen}(\theta+\alpha) \ ][/tex]
A questão nos dá os seguintes complexos :
[tex]\text z_1 = 5.[\ \text{Cos}(\pi) + \text{i.Sen}(\pi) \ ][/tex]
[tex]\displaystyle \text z_2 = 3.[\ \text{Cos}(\frac{\pi}{3}) + \text{i.Sen}(\frac{\pi}{3}) \ ][/tex]
Fazendo o produto dos complexos, temos :
[tex]\displaystyle \text z_1.\text z_2 = 5.3 .[\ \text{Cos}(\pi+\frac{\pi}{3}) + \text{i.Sen}(\pi+ \frac{\pi}{3}) \ ][/tex]
[tex]\boxed{\displaystyle \text z_1.\text z_2 =15 .[\ \text{Cos}(\frac{4.\pi}{3}) + \text{i.Sen}( \frac{4.\pi}{3}) \ ]}\checkmark[/tex]
