Goniometrický tvar komplexního čísla
Uprav zadané komplexní číslo a pak ho zapiš v goniometrickém tvaru:
\large z_3=\left( -4\sqrt {3}-4\text{i}\right) \cdot \Large \frac{\sqrt {3}}{2}\large \left( \cos {\Large \frac{47\pi}{6}\large }+\text{i}\sin {\Large \frac{47\pi}{6}\large }\right)
z_1\cdot z_2=|z_1|\cdot|z_2|\:\left\lbrack\cos\:\left(\alpha_1+\alpha_2\right)+i\:\sin\left(\alpha_1+\alpha_2\right)\right\rbrack
8\left(\cos\:\frac{7\pi}{6}+\:i\:\sin\:\frac{7\pi}{6}\right)\cdot\frac{\sqrt3}{2}\left(\cos\:\frac{47\pi}{6}+i\:\sin\frac{47\pi}{6}\right)=
\large =4\sqrt {3}\left( \cos {\pi}+\text{i}\sin {\pi}\right)
z_1\cdot z_2=|z_1|\cdot|z_2|\:\left\lbrack\cos\:\left(\alpha_1+\alpha_2\right)+i\:\sin\left(\alpha_1+\alpha_2\right)\right\rbrack
8\left(\cos\:\frac{7\pi}{6}+\:i\:\sin\:\frac{7\pi}{6}\right)\cdot\frac{\sqrt3}{2}\left(\cos\:\frac{47\pi}{6}+i\:\sin\frac{47\pi}{6}\right)=
\large =4\sqrt {3}\left( \cos {\frac{4\pi}{3}}+\text{i}\sin {\frac{4\pi}{3}}\right)
z_1\cdot z_2=|z_1|\cdot|z_2|\:\left\lbrack\cos\:\left(\alpha_1+\alpha_2\right)+i\:\sin\left(\alpha_1+\alpha_2\right)\right\rbrack
8\left(\cos\:\frac{7\pi}{6}+\:i\:\sin\:\frac{7\pi}{6}\right)\cdot\frac{\sqrt3}{2}\left(\cos\:\frac{47\pi}{6}+i\:\sin\frac{47\pi}{6}\right)=
\large =4\sqrt {3}\left( \cos {\frac{2\pi}{3}}+\text{i}\sin {\frac{2\pi}{3}}\right)
z_1\cdot z_2=|z_1|\cdot|z_2|\:\left\lbrack\cos\:\left(\alpha_1+\alpha_2\right)+i\:\sin\left(\alpha_1+\alpha_2\right)\right\rbrack
8\left(\cos\:\frac{7\pi}{6}+\:i\:\sin\:\frac{7\pi}{6}\right)\cdot\frac{\sqrt3}{2}\left(\cos\:\frac{47\pi}{6}+i\:\sin\frac{47\pi}{6}\right)=
\large =4\sqrt {3}\left( \cos {\frac{5\pi}{3}}+\text{i}\sin {\frac{5\pi}{3}}\right)