Question Number 18243 by ajfour last updated on 17/Jul/17
Commented by ajfour last updated on 17/Jul/17
![If z=∣z∣(e^(i𝛗) cos θ+jsin θ) z_(𝛗+△𝛗) =∣z∣(e^(i𝛗+i△𝛗) cos θ+jsin θ) z^(θ+△θ) =∣z∣[e^(i𝛗) cos (θ+△θ)+jsin (θ+△θ)] Then prove : (i)z^(θ+△θ) =(cos △θ)z+(sin △θ)(∂z/∂θ) (ii)z_(𝛗+△𝛗) =(cos △𝛗)z+ (sin △𝛗)(∂z/∂𝛗)+j(1−cos △𝛗)sin θ (iii) i^2 =−1 ; find j^2 Prove z_(𝛗+△𝛗) ^(θ+△θ) =(cos △θ)z_(𝛗+△𝛗) + (sin △θ)((∂z/∂θ))_(𝛗+△𝛗) .](https://www.tinkutara.com/question/Q18244.png)
$$\mathrm{If}\:\:\:\:\mathrm{z}=\mid\mathrm{z}\mid\left(\mathrm{e}^{\mathrm{i}\boldsymbol{\phi}} \mathrm{cos}\:\theta+\mathrm{jsin}\:\theta\right) \\ $$$$\mathrm{z}_{\boldsymbol{\phi}+\bigtriangleup\boldsymbol{\phi}} =\mid\mathrm{z}\mid\left(\mathrm{e}^{\mathrm{i}\boldsymbol{\phi}+\mathrm{i}\bigtriangleup\boldsymbol{\phi}} \mathrm{cos}\:\theta+\mathrm{jsin}\:\theta\right) \\ $$$$\mathrm{z}^{\theta+\bigtriangleup\theta} \:=\mid\mathrm{z}\mid\left[\mathrm{e}^{\mathrm{i}\boldsymbol{\phi}} \mathrm{cos}\:\left(\theta+\bigtriangleup\theta\right)+\mathrm{jsin}\:\left(\theta+\bigtriangleup\theta\right)\right] \\ $$$$\mathrm{Then}\:\mathrm{prove}\:: \\ $$$$\:\left(\mathrm{i}\right)\mathrm{z}^{\theta+\bigtriangleup\theta} =\left(\mathrm{cos}\:\bigtriangleup\theta\right)\mathrm{z}+\left(\mathrm{sin}\:\bigtriangleup\theta\right)\frac{\partial\mathrm{z}}{\partial\theta} \\ $$$$\left(\mathrm{ii}\right)\mathrm{z}_{\boldsymbol{\phi}+\bigtriangleup\boldsymbol{\phi}} =\left(\mathrm{cos}\:\bigtriangleup\boldsymbol{\phi}\right)\mathrm{z}+ \\ $$$$\:\:\:\:\left(\mathrm{sin}\:\bigtriangleup\boldsymbol{\phi}\right)\frac{\partial\mathrm{z}}{\partial\boldsymbol{\phi}}+\mathrm{j}\left(\mathrm{1}−\mathrm{cos}\:\bigtriangleup\boldsymbol{\phi}\right)\mathrm{sin}\:\theta \\ $$$$\left(\mathrm{iii}\right)\:\:\mathrm{i}^{\mathrm{2}} =−\mathrm{1}\:\:;\:\:\mathrm{find}\:\:\mathrm{j}^{\mathrm{2}} \\ $$$$\:\mathrm{Prove}\:\mathrm{z}_{\boldsymbol{\phi}+\bigtriangleup\boldsymbol{\phi}} ^{\theta+\bigtriangleup\theta} =\left(\mathrm{cos}\:\bigtriangleup\theta\right)\mathrm{z}_{\boldsymbol{\phi}+\bigtriangleup\boldsymbol{\phi}} + \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{sin}\:\bigtriangleup\theta\right)\left(\frac{\partial\mathrm{z}}{\partial\theta}\right)_{\boldsymbol{\phi}+\bigtriangleup\boldsymbol{\phi}} \:. \\ $$
Commented by ajfour last updated on 09/Jul/18
$$\mathrm{3D}-\mathrm{complex}\:{numbers} \\ $$