Goniometrický tvar komplexního čísla
Uprav zadané komplexní číslo a pak ho zapiš v goniometrickém tvaru:
\large z_4 = \Large \frac{\cos {\frac{15\pi}{4}}+\text{i}\sin {\frac{15\pi}{4}}}{\sqrt {6}-\text{i}\sqrt {6}}\large
\frac{z_1}{z_2}=|\frac{z_1}{z_2}|\:\left\lbrack\cos\left(\alpha_1-\alpha_2\right)-\sin\left(\alpha_1-\alpha_2\right)\right\rbrack
\frac{\cos\:\frac{15\pi}{4}+i\:\sin\:\frac{15\pi}{4}}{2\sqrt3\left(\cos\:\frac{9\pi}{4}+i\:\sin\:\frac{9\pi}{4}\right)}
\large =\Large \frac{\sqrt {3}}{6}\large \left( \cos {\frac{\pi}{2}}+\text{i}\sin {\frac{\pi}{2}}\right)
\frac{z_1}{z_2}=|\frac{z_1}{z_2}|\:\left\lbrack\cos\left(\alpha_1-\alpha_2\right)+\sin\left(\alpha_1-\alpha_2\right)\right\rbrack
\frac{\cos\:\frac{15\pi}{4}+i\:\sin\:\frac{15\pi}{4}}{2\sqrt3\left(\cos\:\frac{11\pi}{4}+i\:\sin\:\frac{11\pi}{4}\right)}
\large =\Large \frac{\sqrt {3}}{6}\large \left( \cos {\frac{\pi}{4}}+\text{i}\sin {\frac{\pi}{4}}\right)
\frac{z_1}{z_2}=|\frac{z_1}{z_2}|\:\left\lbrack\cos\left(\alpha_1-\alpha_2\right)+\sin\left(\alpha_1-\alpha_2\right)\right\rbrack
\frac{\cos\:\frac{15\pi}{4}+i\:\sin\:\frac{15\pi}{4}}{2\sqrt3\left(\cos\:\frac{7\pi}{4}+i\:\sin\:\frac{7\pi}{4}\right)}
\large =\Large \frac{\sqrt {3}}{6}\large \left( \cos {0}+\text{i}\sin {0}\right)
\frac{z_1}{z_2}=|\frac{z_1}{z_2}|\:\left\lbrack\cos\left(\alpha_1+\alpha_2\right)+\sin\left(\alpha_1+\alpha_2\right)\right\rbrack
\frac{\cos\:\frac{15\pi}{4}+i\:\sin\:\frac{15\pi}{4}}{2\sqrt3\left(\cos\:\frac{5\pi}{4}+i\:\sin\:\frac{5\pi}{4}\right)}
\large =\Large \frac{\sqrt {3}}{6}\large \left( \cos {\pi}+\text{i}\sin {\pi}\right)