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Question Number 22769 by Bruce Lee last updated on 22/Oct/17

write to geometry form   (3/5)+(4/5)i    help plz

$$\boldsymbol{\mathrm{write}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{geometry}}\:\boldsymbol{\mathrm{form}} \\ $$$$\:\frac{\mathrm{3}}{\mathrm{5}}+\frac{\mathrm{4}}{\mathrm{5}}\boldsymbol{\mathrm{i}} \\ $$$$ \\ $$$$\boldsymbol{\mathrm{help}}\:\boldsymbol{\mathrm{plz}}\: \\ $$

Answered by $@ty@m last updated on 22/Oct/17

Let (3/5)+(4/5)i=rcos θ+i.rsin θ  ⇒rcos θ=(3/5) & rsin θ=(4/5)  ⇒r=1 & θ=tan^(−1) (4/3)  ∴(3/5)+(4/5)i=cos θ+i.sin θ  where θ=tan^(−1) (4/3)

$${Let}\:\frac{\mathrm{3}}{\mathrm{5}}+\frac{\mathrm{4}}{\mathrm{5}}\boldsymbol{\mathrm{i}}={r}\mathrm{cos}\:\theta+{i}.{r}\mathrm{sin}\:\theta \\ $$$$\Rightarrow{r}\mathrm{cos}\:\theta=\frac{\mathrm{3}}{\mathrm{5}}\:\&\:{r}\mathrm{sin}\:\theta=\frac{\mathrm{4}}{\mathrm{5}} \\ $$$$\Rightarrow{r}=\mathrm{1}\:\&\:\theta=\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{4}}{\mathrm{3}} \\ $$$$\therefore\frac{\mathrm{3}}{\mathrm{5}}+\frac{\mathrm{4}}{\mathrm{5}}\boldsymbol{\mathrm{i}}=\mathrm{cos}\:\theta+{i}.\mathrm{sin}\:\theta \\ $$$${where}\:\theta=\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{4}}{\mathrm{3}} \\ $$

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