Olga musela nájsť algebraický tvar komplexného čísla $$\frac{\cos{\frac{7\pi}{6}}+\mathrm{i}\sin{\frac{7\pi}{6}}}{{2\mathrm{i}}}\cdot\frac{\mathrm{i}-1}{\cos{\frac{5\pi}{3}}+\mathrm{i}\sin{\frac{5\pi}{3}}}.$$
Urobila Olga vo svojom riešení chybu? Ak áno, uveďte v ktorom kroku.
Oľginé riešenie:
$$ \begin{aligned} \frac{\cos{\frac{7\pi}{6}}+\mathrm{i}\sin{\frac{7\pi}{6}}}{{2\mathrm{i}}}\cdot\frac{\mathrm{i}-1}{\cos{\frac{5\pi}{3}}+\mathrm{i}\sin{\frac{5\pi}{3}}} &\stackrel{(1)}= \frac{\cos{\frac{7\pi}{6}}+\mathrm{i}\sin{\frac{7\pi}{6}}}{2\left(\cos\frac{\pi}{2}+\mathrm{i}\sin\frac{\pi}{2}\right)}\cdot\frac{\sqrt{2}\left(\cos\frac{3\pi}{4}+\mathrm{i}\sin\frac{3\pi}{4}\right)}{\cos{\frac{5\pi}{3}}+\mathrm{i}\sin{\frac{5\pi}{3}}}= \cr &\stackrel{(2)}= \frac{\sqrt{2}}{2}\left[\cos\left(\frac{7\pi}{6}+\frac{3\pi}{4}-\frac{\pi}{2}-\frac{5\pi}{3}\right)+ \mathrm{i}\sin\left(\frac{7\pi}{6}+\frac{3\pi}{4}-\frac{\pi}{2}-\frac{5\pi}{3}\right)\right]=\cr &\stackrel{(3)}= \frac{\sqrt{2}}{2}\left[\cos\left(-\frac{\pi}{4}\right)+ \mathrm{i}\sin\left(-\frac{\pi}{4}\right)\right] =\cr&\stackrel{(4)}= \frac{\sqrt{2}}{2}\left(\frac{\sqrt{2}}{2}-\mathrm{i}\frac{\sqrt{2}}{2}\right)=\cr& \stackrel{(5)}= \frac12-\frac12\mathrm{i} \end{aligned} $$
Olga vyriešila príklad správne.
Chyba je v kroku (1). Správne by malo byť:
$$\frac{\cos{\frac{7\pi}{6}}+\mathrm{i}\sin{\frac{7\pi}{6}}}{2\left(\cos\frac{\pi}{2}+\mathrm{i}\sin\frac{\pi}{2}\right)}\cdot\frac{\sqrt{2}\left(\cos\frac{7\pi}{4}+\mathrm{i}\sin\frac{7\pi}{4}\right)}{\cos{\frac{5\pi}{3}}+\mathrm{i}\sin{\frac{5\pi}{3}}}$$
Chyba je v kroku (2). Správne by malo byť:
$$\frac{\sqrt{2}}{2}\left[\cos\left(\frac{\frac{7\pi}{6}\cdot\frac{3\pi}{4}}{\frac{\pi}{2}\cdot\frac{5\pi}{3}}\right)+ \mathrm{i}\sin\left(\frac{\frac{7\pi}{6}\cdot\frac{3\pi}{4}}{\frac{\pi}{2}\cdot\frac{5\pi}{3}}\right)\right]$$
Chyba je v kroku (4). Správne by malo byť:
$$\frac{\sqrt{2}}{2}\left(-\frac{\sqrt{2}}{2}-\mathrm{i}\frac{\sqrt{2}}{2}\right)$$