Convert the following polar equations to rectangular equations. a. r=1/(4sinx-3sinx) b.r=4

Question
Equations
Convert the following polar equations to rectangular equations.
a. $$\displaystyle{r}=\frac{{1}}{{{4}{\sin{{x}}}-{3}{\sin{{x}}}}}$$
b.r=4

2020-11-03
Step 1
To Determine:
Convert the following polar equations to rectangular equations.
a.$$\displaystyle\frac{{1}}{{{4}{\sin{{x}}}-{3}{\sin{{x}}}}}$$
b. r =4
Step 2
Explanation:
a.
Given that,
$$\displaystyle{r}=\frac{{1}}{{{4}{\sin{{x}}}-{3}{\sin{{x}}}}}=\frac{{1}}{{{\sin{{x}}}}}$$
Multiply both the side r
$$\displaystyle{r}^{{2}}=\frac{{r}}{{{\sin{{x}}}}}$$
We know that,
$$\displaystyle{r}^{{2}}={x}^{{2}}+{y}^{{2}}$$
$$\displaystyle{x}^{{2}}+{y}^{{2}}=\frac{{r}}{{{\sin{{x}}}}}$$
and we know that,
$$\displaystyle{y}={r}{\sin{{x}}}$$
$$\displaystyle{r}=\frac{{y}}{{{\sin{{x}}}}}$$
Put the value of r
$$\displaystyle{x}^{{2}}+{y}^{{2}}=\frac{{y}}{{{\sin{{x}}}}}/{\left({\sin{{x}}}\right)}$$
$$\displaystyle{x}^{{2}}+{y}^{{2}}=\frac{{y}}{{{{\sin}^{{2}}{x}}}}$$

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