Solution:
Suppose we have complex numbers z₁ and z₂ expressed as
[tex]\begin{gathered} z_1=r_1(\cos \theta_1+isin\theta_1) \\ z_2=r_2(\cos \theta_2+isin\theta_2) \end{gathered}[/tex]
Thus, z₁z₂ is evaluated as
[tex]z_1z_2=r_1r_2(\cos (\theta_1+\theta_2)+i\sin (\theta_1+\theta_2)[/tex]
Given the complex numbers z₁ and z₂ as expressed below:
[tex]\begin{gathered} z_1=12_{}(\cos 30\degree+i\sin 30\degree_{}) \\ z_2=4_{}(\cos 240\degree+i\sin 240\degree_{}) \end{gathered}[/tex]
Then to evaluate the product of the complex numbers, we have
[tex]\begin{gathered} z_1z_2=r_1r_2(\cos (\theta_1+\theta_2)+i\sin (\theta_1+\theta_2) \\ \text{where} \\ r_1=12 \\ r_2=4 \\ \theta_1=30 \\ \theta_2=240 \\ \text{thus,} \\ z_1z_2=12\times4_{}(\cos (30_{}+240_{})+i\sin (30_{}+240)) \\ \Rightarrow48(\cos 270\degree+i\sin 270^{}\degree) \end{gathered}[/tex]
Hence, z₁z₂ is evaluated to be
[tex]48(\cos 270\degree+i\sin 270^{}\degree)[/tex]
The first option is the correct answer.
