קיץ 2021 · מועד א

$z^4 = -16 = 16\,\text{cis}\,180°$

$r^4 = 16 \;\Rightarrow\; r = 2$

$4\theta = 180° + 360°k \;\Rightarrow\; \theta = 45° + 90°k, \quad k = 0,1,2,3$

$z_1 = 2\,\text{cis}\,45° = \sqrt 2 + \sqrt 2\,i$

$z_2 = 2\,\text{cis}\,135° = -\sqrt 2 + \sqrt 2\,i$

$z_3 = 2\,\text{cis}\,225° = -\sqrt 2 - \sqrt 2\,i$

$z_4 = 2\,\text{cis}\,315° = \sqrt 2 - \sqrt 2\,i$

$z_1 \cdot \text{cis}\,45° = 2\,\text{cis}\,90° = 2i \;\Rightarrow\; (0,\,2)$

$z_2 \cdot \text{cis}\,45° = 2\,\text{cis}\,180° = -2 \;\Rightarrow\; (-2,\,0)$

$z_3 \cdot \text{cis}\,45° = 2\,\text{cis}\,270° = -2i \;\Rightarrow\; (0,\,-2)$

$z_4 \cdot \text{cis}\,45° = 2\,\text{cis}\,360° = 2 \;\Rightarrow\; (2,\,0)$

$c = 2^{16}\,\text{cis}\,0° = 65536$

$S = 16 \cdot 2\sin 22.5° = 32\sin 22.5°$

$\sin 22.5° = \dfrac{\sqrt{2 - \sqrt 2}}{2} \;\Rightarrow\; S = 16\sqrt{2 - \sqrt 2} \approx 12.25$