It sounds like you're saying
[tex]z = \dfrac38 \left(\cos\left(\dfrac\pi8\right) + i \sin\left(\dfrac\pi8\right)\right)[/tex]
[tex]w = 2 \left(\cos\left(\dfrac\pi{16}\right) + i \sin\left(\dfrac\pi{16}\right)\right)[/tex]
The product [tex]zw[/tex] is obtained by multiplying the moduli and adding the arguments. In other words
[tex]z = |z| e^{i\arg(z)} \text{ and } w = |w| e^{i\arg(w)} \implies zw = |z||w| e^{i(\arg(z)+\arg(w))}[/tex]
where [tex]e^{it}=\cos(t)+i\sin(t)[/tex], so that
[tex]zw = \dfrac38\times2 \left(\cos\left(\dfrac\pi8+\dfrac\pi{16}\right) + i \sin\left(\dfrac\pi8 + \dfrac\pi{16}\right)\right) = \boxed{\dfrac34 \left(\cos\left(\dfrac{3\pi}{16}\right) + i \sin\left(\dfrac{3\pi}{16}\right)\right)}[/tex]
