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Question Number 7582 by A WLAN last updated on 04/Sep/16

Commented by A WLAN last updated on 04/Sep/16

$$Howtosolvethisproblem?$$

Answered by Yozzia last updated on 04/Sep/16

$$w=1/z.Youcanplottheinitialpointz$$$$andthepointwonanArganddiagramtodeducetheangleof$$$$rotation.$$$$z=1\Rightarrow w=1/1=z\Rightarrow angleofrotationis0°$$$$abouttheorigin.z=1isinvariantunderthetransformation.$$$$z=i\Rightarrow w=1/i=-i=-z\Rightarrow angleofrotationis180°$$$$abouttheorigin.$$$$------------------------------$$$$Alternatively,wecanfindtheargument$$$$of\frac{w}{z}.Letw=r_1{e}^{i\varphi }andz=r_2{e}^{i\theta }.$$$$\frac{w}{z}=\frac{r_1}{r_2}{e}^{i(\varphi -\theta )}\Rightarrow arg\left(w/z\right)=\varphi -\theta =arg\left(w\right)-arg\left(z\right)$$$$Forz=1\Rightarrow arg\left(z\right)=\theta ={tan}^{-1}\frac{0}{1}=0radians$$$$\because w=1/z\Rightarrow w=1\Rightarrow arg\left(w\right)=\varphi ={tan}^{-1}\frac{0}{1}=0radians$$$$\therefore arg\left(w/z\right)=0-0=0radians.$$$$\Rightarrow Theanglebetweenwandzis0radians.$$$$\Rightarrow Theangleofrotationabouttheoriginis0radians.$$$$z=i\Rightarrow arg\left(z\right)=\theta =\underset{x\to 0^{+}}{\lim }{tan}^{-1}\frac{1}{x}=\frac{\pi }{2}$$$$w=1/z=1/i=-i\Rightarrow arg\left(w\right)=\varphi =\underset{x\to 0^{+}}{\lim }tan\frac{-1}{x}=-\underset{x\to 0^{+}}{\lim }{tan}^{-1}\frac{1}{x}=\frac{-\pi }{2}$$$$\therefore arg\left(w/z\right)=\frac{-\pi }{2}-\frac{\pi }{2}=-\pi .$$$$\Rightarrow Theanglebetweenwandzis\pi radians.$$$$\Rightarrow Theangleofrotationis\pi radiansabouttheorigin.$$$$$$

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