Question

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

Exponential models
Find the Polar, General Polar, Exponential and General Exponential Form of $$\displaystyle-{5}+{9}{j}$$

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
$$\displaystyle{z}={e}{r}^{{{i}\theta}}$$
$$\displaystyle{z}=\sqrt{{{106}}}{e}^{{{i}{{\tan}^{{-{1}}}{\left({\frac{{-{9}}}{{{5}}}}\right)}}}}$$