Chapter 8.1, Problem 57E

To determine

To convert the given polar equation to rectangular coordinates

  $r=1+\cos \theta$

Answer to Problem 57E

The rectangular equation is $x^2+y^2=\sqrt{x^2+y^2}+x$ .

Explanation of Solution

Given: Polar equation: $r=1+\cos \theta$

Formula Used:

A polar equation is any equation that describes a relation between r and θ , where r represents the distance from the pole (origin) to a point on a curve, and θ represents the counter-clockwise angle made by a point on a curve, the pole, and the positive x-axis.

Also, $x=r\cos \theta$ and $y=r\sin \theta$

  $r=\sqrt{x^2+y^2}$ and $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

Calculation:

Given Polar equation is $r=1+\cos \theta$

Using the above definition, we have:

  $r=1+\cos \theta$ ;

Multiplying both the sides by $r$ , we have:

  $r\times r=\left(1+\cos \theta \right)\times r$

  $\begin{array}{l}{r}^2=r+\left(r\cos \theta \right)\\ x^2+y^2=\sqrt{x^2+y^2}+x\end{array}$

Thus, the rectangular equation is $x^2+y^2=\sqrt{x^2+y^2}+x$ .

Conclusion:

The rectangular equation is $x^2+y^2=\sqrt{x^2+y^2}+x$ .