קיץ 2023 · מועד א

$z^3 = \dfrac{1}{z^3} \Rightarrow z^6 = 1$

$z^6 = \text{cis}(0° + 360°k)$

$z_k = \text{cis}\left(\dfrac{360°k}{6}\right) = \text{cis}(60°k)$, $\;k=0,1,2,3,4,5$

$4.5 < k < 6 \Rightarrow k = 5$

$z_0 = \text{cis}(300°) = \tfrac12 - \tfrac{\sqrt3}{2}i$

$A = d z_0 = d\,\text{cis}(300°)$

$B = d\,i\,z_0 = d\,\text{cis}(90°)\,\text{cis}(300°) = d\,\text{cis}(390°) = d\,\text{cis}(30°)$

$C = d(z_0)^4 = d\,\text{cis}(4\cdot 300°) = d\,\text{cis}(1200°) = d\,\text{cis}(120°)$

$S_{ABC} = \tfrac12\,AC\cdot OB = \tfrac12\cdot 2d\cdot d = d^2$

$d^2 = 5d + 6$

$d^2 - 5d - 6 = 0 \Rightarrow (d-6)(d+1) = 0$

$d > 0 \Rightarrow d = 6$

$(z_0)^2 = (\text{cis}300°)^2 = \text{cis}(600°) = \text{cis}(240°)$

$\dfrac{1}{(z_0)^2} = \text{cis}(-240°) = \overline{\text{cis}(240°)}$

$(z_0)^2 - \dfrac{1}{(z_0)^2} = \text{cis}(240°) - \overline{\text{cis}(240°)} = 2i\,\text{Im}(\text{cis}240°)$

$= 2i\sin(240°) = 2i\cdot\left(-\tfrac{\sqrt3}{2}\right) = -\sqrt3\,i$

$1 + i = \sqrt2\,\text{cis}(45°)$

$w = (-\sqrt3\,i)\cdot\sqrt2\,\text{cis}(45°) = \sqrt3\,\text{cis}(270°)\cdot\sqrt2\,\text{cis}(45°)$

$w = \sqrt6\,\text{cis}(315°) = \sqrt6\,\text{cis}(-45°)$

$w^n = (\sqrt6)^n\,\text{cis}(-45n°) = 6^{n/2}\,\text{cis}(-45n°)$

$6^{n/2} > 6^1$

$n > 2$

$-45n° = 90° + 180°k \Rightarrow n = -2 - 4k$