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

\(\displaystyle{x}^{{2}}+{y}^{{2}}=\frac{{y}}{{{\sin{{x}}}}}/{\left({\sin{{x}}}\right)}\)

\(\displaystyle{x}^{{2}}+{y}^{{2}}=\frac{{y}}{{{{\sin}^{{2}}{x}}}}\)

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

\(\displaystyle{x}^{{2}}+{y}^{{2}}=\frac{{y}}{{{\sin{{x}}}}}/{\left({\sin{{x}}}\right)}\)

\(\displaystyle{x}^{{2}}+{y}^{{2}}=\frac{{y}}{{{{\sin}^{{2}}{x}}}}\)