Question

Find the Polar, General Polar, Exponential and General Exponential Form of -5+9j

Exponential models
ANSWERED
asked 2021-01-04
Find the Polar, General Polar, Exponential and General Exponential Form of \(\displaystyle-{5}+{9}{j}\)

Answers (1)

2021-01-05
Given,
\(\displaystyle{z}=-{5}+{9}{i}\)
where,
\(\displaystyle{r}{\cos{{\left(\theta\right)}}}=-{5}\) (1)
\(\displaystyle{r}{\sin{{\left(\theta\right)}}}={9}\) (2)
\(\displaystyle{r}^{{2}}{\left({{\cos}^{{2}}{\left(\theta\right)}}+{{\sin}^{{2}}{\left(\theta\right)}}\right)}={25}+{81}\)
\(\displaystyle{r}^{{2}}={106}\)
\(\displaystyle{r}=\sqrt{{{106}}}\)
From equation (1) and (2)
\(\displaystyle{\frac{{{\sin{{\left(\theta\right)}}}}}{{{\cos{{\left(\theta\right)}}}}}}=-{\frac{{{9}}}{{{5}}}}\)
\(\displaystyle{\tan{{\left(\theta\right)}}}={\frac{{-{9}}}{{{5}}}}\)
polar form of number
\(\displaystyle{z}={r}{\left({\cos{{\left(\theta\right)}}}+{i}{\sin{{\left(\theta\right)}}}\right.}\)
Put values in equation
\(\displaystyle{z}=\sqrt{{{106}}}{\left({\cos{{\left({\tan{{\left\lbrace-{1}\right\rbrace}}}{\left(-{\frac{{{9}}}{{{5}}}}\right)}\right)}}}+{i}{\sin{{\left({{\tan}^{{-{1}}}{\left({\frac{{-{9}}}{{{5}}}}\right)}}\right)}}}\right)}\)
Exponential form
\(\displaystyle{z}={e}{r}^{{{i}\theta}}\)
\(\displaystyle{z}=\sqrt{{{106}}}{e}^{{{i}{{\tan}^{{-{1}}}{\left({\frac{{-{9}}}{{{5}}}}\right)}}}}\)
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