$\frac{-\frac{\sqrt3}{2}-\frac12\mathrm{i}}{{2\mathrm{i}}}\cdot\frac{\mathrm{i}-1}{\cos{\frac{5\pi}{3}}+\mathrm{i}\sin{\frac{5\pi}{3}}}$

Project ID: 
3000000015
SubArea: 
Algebrický a goniometrický tvar komplexného čísla
Question: 

Eugen musel nájsť algebraický tvar komplexného čísla $$\frac{-\frac{\sqrt3}{2}-\frac12\mathrm{i}}{{2\mathrm{i}}}\cdot\frac{\mathrm{i}-1}{\cos{\frac{5\pi}{3}}+\mathrm{i}\sin{\frac{5\pi}{3}}}.$$

Urobil Eugen vo svojom riešení chybu? Ak áno, uveďte v ktorom kroku.

Eugenovo riešenie:

$$ \begin{aligned} \frac{-\frac{\sqrt3}{2}-\frac12\mathrm{i}}{{2\mathrm{i}}}\cdot\frac{\mathrm{i}-1}{\cos{\frac{5\pi}{3}}+\mathrm{i}\sin{\frac{5\pi}{3}}} &\stackrel{(1)}= \frac{-\frac{\sqrt3}{2}-\frac12\mathrm{i}}{2\mathrm{i}}\cdot\frac{\mathrm{i}-1}{\frac12-\frac{\sqrt{3}}{2}\mathrm{i}}= \cr &\stackrel{(2)}= \frac{-\frac{\sqrt3}{2}\mathrm{i}+\frac12+\frac{\sqrt3}{2}+\frac12\mathrm{i}}{\mathrm{i}+\sqrt3} =\cr &\stackrel{(3)}= \frac12\cdot\frac{1+\sqrt3-(1+\sqrt3)\mathrm{i}}{\sqrt3+\mathrm{i}} =\cr&\stackrel{(4)}= \frac12\cdot\frac{\left[1+\sqrt3-(1+\sqrt3)\mathrm{i}\right](\sqrt3-\mathrm{i})}{(\sqrt3+\mathrm{i})(\sqrt3-\mathrm{i})} =\cr& \stackrel{(5)}= \frac{\sqrt3-\mathrm{i}+3-\sqrt3 \mathrm{i}-\sqrt3 \mathrm{i}-1-3\mathrm{i}-\sqrt3}{8} =\cr& \stackrel{(6)}= \frac{2-(4+2\sqrt3)\mathrm{i}}{8} =\cr& \stackrel{(7)}= \frac14-\frac{2+\sqrt3}{4}\mathrm{i} \end{aligned} $$

Answer 1: 

Eugen vyriešil príklad správne.

Answer 2: 

Chyba je v kroku (3). Správne by malo byť:

$$ \frac12\cdot\frac{1+\sqrt3+(1-\sqrt3)\mathrm{i}}{\sqrt3+\mathrm{i}} $$

Answer 3: 

Chyba je v kroku (5). Správne by malo byť:

$$\frac{\sqrt3+3-\sqrt3\mathrm{i}-3\mathrm{i}-\mathrm{i}-\sqrt3\mathrm{i}-1-\sqrt3}{8}$$

Answer 4: 

Chyba je v kroku (7). Správne by malo byť:

$$\frac14-\frac{2-\sqrt3}{4}\mathrm{i}$$

Fixed Answer: 
All Fixed
Correct Answer: 
Answer 2
Hint: 

Eugenovo riešenie s opravenou chybou:

$$ \begin{aligned} \frac{-\frac{\sqrt3}{2}-\frac12\mathrm{i}}{{2\mathrm{i}}}\cdot\frac{\mathrm{i}-1}{\cos{\frac{5\pi}{3}}+\mathrm{i}\sin{\frac{5\pi}{3}}} &\stackrel{(1)}= \frac{-\frac{\sqrt3}{2}-\frac12\mathrm{i}}{2\mathrm{i}}\cdot\frac{\mathrm{i}-1}{\frac12-\frac{\sqrt{3}}{2}\mathrm{i}}= \cr &\stackrel{(2)}= \frac{-\frac{\sqrt3}{2}\mathrm{i}+\frac12+\frac{\sqrt3}{2}+\frac12\mathrm{i}}{\mathrm{i}+\sqrt3} =\cr &\stackrel{(3)}= \frac12\cdot\frac{1+\sqrt3+(1-\sqrt3)\mathrm{i}}{\sqrt3+\mathrm{i}} =\cr&\stackrel{(4)}= \frac12\cdot\frac{\left[1+\sqrt3+(1-\sqrt3)\mathrm{i}\right](\sqrt3-\mathrm{i})}{(\sqrt3+\mathrm{i})(\sqrt3-\mathrm{i})} =\cr& \stackrel{(5)}= \frac{\sqrt3-\mathrm{i}+3-\sqrt3 \mathrm{i}+\sqrt3 \mathrm{i}+1-3\mathrm{i}-\sqrt3}{8} =\cr& \stackrel{(6)}= \frac{-\mathrm{i}+3+1-3\mathrm{i}}{8} =\cr& \stackrel{(7)}= \frac{4-4\mathrm{i}}{8} =\cr& \stackrel{(8)}= \frac12-\frac12\mathrm{i} \end{aligned} $$