Step 1

Given polar equation is:

\(r\sin(0-\frac{\pi}{4})=2\)

Step 2

To convert polar equation to a rectangular equation,

put \(r=\sqrt{x^{2}+y^{2}}\ and\ 0=\frac{y}{x}\)

\(\sqrt{x^{2}+y^{2}} \sin (\tan^{-1}(\frac{y}{x})-\frac{\pi}{4})=2\)

\(\sin(\tan^{-1}(\frac{y}{x})-\frac{\pi}{4})=\frac{2}{\sqrt{x^{2}+y^{2}}}\)

\(\tan^{-1}(\frac{y}{x})-\frac{\pi}{4}=\sin^{-1}(\frac{2}{\sqrt{x^{2}+y^{2}}})\)

\(\tan^{-1}(\frac{y}{x})=\sin^{-1}(\frac{2}{\sqrt{x^{2}+y^{2}}})+\frac{\pi}{4}\)

Given polar equation is:

\(r\sin(0-\frac{\pi}{4})=2\)

Step 2

To convert polar equation to a rectangular equation,

put \(r=\sqrt{x^{2}+y^{2}}\ and\ 0=\frac{y}{x}\)

\(\sqrt{x^{2}+y^{2}} \sin (\tan^{-1}(\frac{y}{x})-\frac{\pi}{4})=2\)

\(\sin(\tan^{-1}(\frac{y}{x})-\frac{\pi}{4})=\frac{2}{\sqrt{x^{2}+y^{2}}}\)

\(\tan^{-1}(\frac{y}{x})-\frac{\pi}{4}=\sin^{-1}(\frac{2}{\sqrt{x^{2}+y^{2}}})\)

\(\tan^{-1}(\frac{y}{x})=\sin^{-1}(\frac{2}{\sqrt{x^{2}+y^{2}}})+\frac{\pi}{4}\)