Goniometrický tvar komplexního čísla
Uprav zadané komplexní číslo a pak ho zapiš v goniometrickém tvaru:
\large z_2=\left( -\sqrt {3}+3\text{i}\right) \cdot 2\left[ \cos {\left( -\Large \frac{3\pi}{4}\large \right) }+\text{i}\sin {\left( -\Large \frac{3\pi}{4}\large \right) }\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{2\pi}{3}+\:i\:\sin\:\frac{2\pi}{3}\right)\cdot\frac{\sqrt3}{2}\left(\cos\:\left(-\frac{3\pi}{4}\right)+i\:\sin\left(-\frac{3\pi}{4}\right)\right.={}
\large =4\sqrt {3}\left[ \cos {\left( -\Large \frac{\pi}{8}\large \right) }+\text{i}\sin {\left( -\Large \frac{\pi}{8}\large \right) }\right] = 4\sqrt {3}\left( \cos {\Large \frac{23\pi}{8}\large }+\text{i}\sin {\Large \frac{23\pi}{8}\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{2\pi}{3}+\:i\:\sin\:\frac{2\pi}{3}\right)\cdot\frac{\sqrt3}{2}\left(\cos\:\left(-\frac{3\pi}{4}\right)+i\:\sin\left(-\frac{3\pi}{4}\right)\right.={}
\large =4\sqrt {3}\left[ \cos {\left( -\Large \frac{\pi}{10}\large \right) }+\text{i}\sin {\left( -\Large \frac{\pi}{10}\large \right) }\right] = 4\sqrt {3}\left( \cos {\Large \frac{23\pi}{10}\large }+\text{i}\sin {\Large \frac{23\pi}{10}\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{2\pi}{3}+\:i\:\sin\:\frac{2\pi}{3}\right)\cdot\frac{\sqrt3}{2}\left(\cos\:\left(-\frac{3\pi}{4}\right)+i\:\sin\left(-\frac{3\pi}{4}\right)\right.={}
\large =4\sqrt {3}\left[ \cos {\left( -\Large \frac{\pi}{12}\large \right) }+\text{i}\sin {\left( -\Large \frac{\pi}{12}\large \right) }\right] = 4\sqrt {3}\left( \cos {\Large \frac{23\pi}{12}\large }+\text{i}\sin {\Large \frac{23\pi}{12}\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{2\pi}{3}+\:i\:\sin\:\frac{2\pi}{3}\right)\cdot\frac{\sqrt3}{2}\left(\cos\:\left(-\frac{3\pi}{4}\right)+i\:\sin\left(-\frac{3\pi}{4}\right)\right.={}
\large =4\sqrt {3}\left[ \cos {\left( -\Large \frac{\pi}{6}\large \right) }+\text{i}\sin {\left( -\Large \frac{\pi}{6}\large \right) }\right] = 4\sqrt {3}\left( \cos {\Large \frac{23\pi}{6}\large }+\text{i}\sin {\Large \frac{23\pi}{6}\large }\right)
Aby šlo komplexní čísla vynásobit, musí obě být goniometrickém tvaru. Poté použiješ vzoreček pro násobení dvou komplexních čísel a je to.