Question

Să se rezolve in C ecuatiile : ex 3 va rog

Answer 1

$e^\Big{i\pi}+1 = 0 \Rightarrow e^\Big{i\pi} = -1 \Rightarrow e^\Big{i\frac{\pi}{2}} = \sqrt{-1} = i\\\ \\ z^5 = -i \Rightarrow z^5 = -e^\Big{i\frac{\pi}{2}} \Rightarrow z^5 = -e^\Big{i(\frac{\pi}{2}+2k\pi)} \Rightarrow z = -e^\Big{i(\frac{\pi}{10}+2k\frac{\pi}{5})}\\ \\ k = 0 \Rightarrow z_1 = -e^\Big{i\frac{\pi}{10}} \\ k=1\Rightarrow z_2 = -e^\Big{i\frac{5\pi}{10}} = -e^\Big{-\frac{\pi}{2}} \Rightarrow z_2 = -i\\k=2 \Rightarrow z_3 = -e^\Big{i\frac{9\pi}{10}}$

$k=3\Rightarrow z_4 = -e^\Big{i\frac{13\pi}{10}}\\ k= 4 \Rightarrow z_5 = -e^\Big{i\frac{17\pi}{10} }$