Chapter 11, Problem 27RE

To determine

To determine: The equivalent Cartesian equation which replaces the polar equation.

Answer to Problem 27RE

The required Cartesian equation is $x=y$ .

Explanation of Solution

Given information:

The polar equation : $r\cos \theta =r\sin \theta$

Calculation:

The given polar equation is $r\cos \theta =r\sin \theta$ .

To replace the polar equation to a Cartesian equation, use the polar-rectangular equations and formulas $x=r\cos \theta$ , $y=r\sin \theta$ , $r^2=x^2+y^2$ and $\tan \theta =\frac{y}{x}$ .

So, $\begin{array}{c}\frac{\cancel{r\cos \theta }}{\cancel{r\cos \theta }}=\frac{\cancel{r}\sin \theta }{\cancel{r}\cos \theta }\\ \tan \theta =1\\ \frac{y}{x}=1\\ x=y\end{array}$

Hence, the required Cartesian equation is $x=y$ .