Задача 45445 отметить первые 5 членов...

gosha_

2020-03-27 18:12:44

отметить первые 5 членов последовательности на комплексной плоскости
z_(n)=(1/(sqrt(3)-i)^(n^2)

sova

2020-03-27 18:38:28

[m]z_{1}=\frac{1}{\sqrt{3}-i}=\frac{\sqrt{3}+i}{4}=\frac{\sqrt{3}}{4}+i\cdot \frac{1}{4}[/m]

точка[m] (\frac{\sqrt{3}}{4};\frac{1}{4})[/m]

[m]z_{2}=\frac{1}{(\sqrt{3}-i)^4}=\frac{1}{((\sqrt{3}-i)^2)^2}=\frac{1}{(2-2\sqrt{3}i)^2}=\frac{1}{4}\cdot \frac{1}{(-2-2\sqrt{3}\cdot i)}=[/m][m]=-\frac{1}{8}\cdot \frac{1}{(1+\sqrt{3}\cdot i)}=-\frac{1}{8}\cdot \frac{1-\sqrt{3}\cdot i}{(1-3)}=-\frac{1}{16}+i\cdot \frac{\sqrt{3}}{16}[/m]

точка[m] (-\frac{1}{16}; \frac{\sqrt{3}}{16})[/m]

и так далее
[m]z_{3}=\frac{1}{(\sqrt{3}-i)^9}[/m]

Можно формулу Муавра применить для возведения в степень

[m]z=|z|(cos\phi+isin \phi)[/m]

[m]z^{n}=|z|^{n}(\cdot(cos(n\phi)+isin (n\phi))[/m]

[m]z=\sqrt{3}-i[/m]

[m]|z|=\sqrt{(\sqrt{3})^2+1^2)}=2[/m]

[m]cos \phi=\frac{x}{|z|}=\frac{\sqrt{3}}{2}[/m]
[m]sin \phi=\frac{y}{|z|}=-\frac{1}{2}[/m]

[m]\phi=-\frac{\pi}{6}[/m]

[m]\sqrt{3}-i=2\cdot (cos(-\frac{\pi}{6})+isin(-\frac{\pi}{6}))[/m]

[m](\sqrt{3}-i)^{4}=2^{4}\cdot (cos(-\frac{4\pi}{6})+isin(-\frac{4\pi}{6}))=2^{4}\cdot(-\frac{1}{2}-i\frac{\sqrt{3}}{2})[/m]

[m](\sqrt{3}-i)^{9}=2^{9}\cdot (cos(-\frac{9\pi}{6})+isin(-\frac{9\pi}{6}))=[/m][m]=2^{9}\cdot (cos(-\frac{3\pi}{2})+isin(-\frac{3\pi}{2}))=-i\cdot 2^{9}[/m]

[m](\sqrt{3}-i)^{16}=2^{16}\cdot (cos(-\frac{16\pi}{6})+isin(-\frac{16\pi}{6}))=2^{16}\cdot(-\frac{1}{2}-i\frac{\sqrt{3}}{2})[/m]

[m](\sqrt{3}-i)^{25}=2^{25}\cdot (cos(-\frac{25\pi}{6})+isin(-\frac{25\pi}{6}))=2^{25}\cdot (\frac{\sqrt{3}}{2}-\frac{1}{2})[/m]