Question and Answers Forum

All Questions      Topic List

Set Theory Questions

Previous in All Question      Next in All Question      

Previous in Set Theory      Next in Set Theory      

Question Number 165616 by amin96 last updated on 05/Feb/22

Commented by mkam last updated on 06/Feb/22

$$\mathit{f}\mathit{i}\mathit{n}\mathit{d}\mathit{a}\mathit{l}\mathit{l}\mathit{t}\mathit{h}\mathit{e}\mathit{r}\mathit{o}\mathit{o}\mathit{t}\mathit{o}\mathit{f}({\mathit{x}}^9-1=0)?$$$$$$$$\subset \mathit{s}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\supset$$$$$$$${\mathit{x}}^9=1\Rightarrow \mathit{x}={\left(1\right)}^{\frac{1}{9}}\Rightarrow {\mathit{x}}_{\mathit{k}}={\mathit{r}}^{\mathit{n}}(\mathit{c}\mathit{o}\mathit{s}\frac{\mathit{\theta }+2\mathit{k}\mathit{\pi }}{\mathit{n}}+\mathit{i}\mathit{s}\mathit{i}\mathit{n}\frac{\mathit{\theta }+2\mathit{k}\mathit{\pi }}{\mathit{n}})$$$$$$$$\mathit{p}\mathit{u}\mathit{t}:\mathit{r}=\sqrt{{\mathit{x}}^2+{\mathit{y}}^2}=\sqrt{{\left(1\right)}^2+{\left(0\right)}^2}=\sqrt{1}=1$$$$$$$$\mathit{\theta }={\mathit{t}\mathit{a}\mathit{n}}^{-1}\left(\frac{\mathit{y}}{\mathit{x}}\right)={\mathit{t}\mathit{a}\mathit{n}}^{-1}\left(\frac{0}{1}\right)={\mathit{t}\mathit{a}\mathit{n}}^{-1}\left(0\right)=0$$$$$$$$\mathit{n}=9,\mathit{k}=0,1,2,........,8$$$$$$$$\mathit{k}=0$$$${\mathit{x}}_0=\mathit{c}\mathit{o}\mathit{s}\left(0\right)+\mathit{i}\mathit{s}\mathit{i}\mathit{n}\left(0\right)=1$$$$\mathit{k}=1$$$${\mathit{x}}_1=\mathit{c}\mathit{o}\mathit{s}\frac{2\mathit{\pi }}{9}+\mathit{i}\mathit{s}\mathit{i}\mathit{n}\frac{2\mathit{\pi }}{9}$$$$\mathit{k}=2$$$${\mathit{x}}_2=\mathit{c}\mathit{o}\mathit{s}\frac{4\mathit{\pi }}{9}+\mathit{i}\mathit{s}\mathit{i}\mathit{n}\frac{4\mathit{\pi }}{9}$$$$\mathit{k}=3$$$${\mathit{x}}_3=\mathit{c}\mathit{o}\mathit{s}\frac{2\mathit{\pi }}{3}+\mathit{i}\mathit{s}\mathit{i}\mathit{n}\frac{2\mathit{\pi }}{3}=-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathit{i}$$$$\mathit{k}=4$$$${\mathit{x}}_4=\mathit{c}\mathit{o}\mathit{s}\frac{8\mathit{\pi }}{9}+\mathit{i}\mathit{s}\mathit{i}\mathit{n}\frac{8\mathit{\pi }}{9}$$$$\mathit{k}=5$$$${\mathit{x}}_5=\mathit{c}\mathit{o}\mathit{s}\frac{10\mathit{\pi }}{9}+\mathit{i}\mathit{s}\mathit{i}\mathit{n}\frac{10\mathit{\pi }}{9}$$$$\mathit{k}=6$$$${\mathit{x}}_6=\mathit{c}\mathit{o}\mathit{s}\frac{4\mathit{\pi }}{3}+\mathit{i}\mathit{s}\mathit{i}\mathit{n}\frac{4\mathit{\pi }}{3}=-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathit{i}$$$$\mathit{k}=7$$$${\mathit{x}}_7=\mathit{c}\mathit{o}\mathit{s}\frac{14\mathit{\pi }}{9}+\mathit{i}\mathit{s}\mathit{i}\mathit{n}\frac{14\mathit{\pi }}{9}$$$$\mathit{k}=8$$$${\mathit{x}}_8=\mathit{c}\mathit{o}\mathit{s}\frac{16\mathit{\pi }}{9}+\mathit{i}\mathit{s}\mathit{i}\mathit{n}\frac{16\mathit{\pi }}{9}$$$$$$$$\left(\mathit{m}\mathit{o}\mathit{h}\mathit{a}\mathit{m}\mathit{m}\mathit{a}\mathit{d}\mathit{a}\mathit{l}\mathit{d}\mathit{o}\mathit{l}\mathit{i}\mathit{m}\mathit{y}\right)$$

Answered by Ar Brandon last updated on 05/Feb/22

$$x^9-1=0\Rightarrow x^9=e^{2\pi ik}\Rightarrow x=e^{\frac{2k}{9}i\pi },k\in [0,8]$$

Answered by alephzero last updated on 05/Feb/22

$$x^9-1=0$$$$x^9=1$$$$x=1$$

Commented by mkam last updated on 06/Feb/22

$$false$$

Commented by alephzero last updated on 11/Feb/22

$$Why?Thisisjust\mathit{o}\mathit{n}\mathit{e}ofall$$$$solutions.$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com