Chapter 8.2, Problem 34E

To determine

To sketch the graph of the given polar equation and express the equation in rectangular coordinates −

  $r=2+\sin \theta$

Answer to Problem 34E

The graph of polar equation is

  

The equation in rectangular coordinates is $\left(x^2+y^2\right)=2\sqrt{\left(x^2+y^2\right)}+y$ .

Explanation of Solution

Given: Polar equation: $r=2+\sin \theta$

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 $r=2+\sin \theta$

Drawing the polar-form curve for $r=2+\sin \theta$ on a graph as per the above definition of polar equation, we have:

  

Now, to express the equation in rectangular coordinates, we have:

  $r=2+\sin \theta$

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

  $\begin{array}{l}r\times r=\left(2+\sin \theta \right)\times r\\ r^2=2r+\left(r\sin \theta \right)\end{array}$

From the above given formulas, we have:

  $\left(x^2+y^2\right)=2\sqrt{\left(x^2+y^2\right)}+y$

Conclusion:

Hence, the graph of polar equation is

  

The equation in rectangular coordinates is $\left(x^2+y^2\right)=2\sqrt{\left(x^2+y^2\right)}+y$ .