Chapter 9.2, Problem 65AYU

In Problems 63-68, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph.

$r=1+\sin \theta ;r=1+\cos \theta$

To determine

To find: Graph the polar equations $r = 1 + \text{sin} \theta ; r = 1 + \text{cos} \theta$ on the same polar grid and find the point of intersections if any.

Answer to Problem 65AYU

Solution

The graph of the polar equations $r = 1 + \text{sin} \theta ; r = 1 + \text{cos} \theta$ are the equations of cardioids.

From the above graph, the point of intersections are $\left(1.207, 1.207\right) \left(0, 0\right) \text{and} \left(- 1.207, - 1.207\right)$.

Given:

It is asked to graph the polar equations $r = 1 + \text{sin} \theta$; $r = 1 + \text{cos} \theta$.

Explanation of Solution

Formula to be used:

Cardioids are characterized by equations of the form

$\begin{array}{lll}r\hfill & =\hfill & a\left(1 \pm \text{cos} \theta \right)\hfill \\ r\hfill & =\hfill & a\left(1 \pm \text{sin} \theta \right)\hfill \end{array}$

where $a > 0$. The graph of a cardioid passes through the pole.

Consider $r = 1 + \text{sin} \theta$,

This polar equation can be written as

$r = 1 \left(1 + \text{sin} \theta \right)$

The above equation is an equation of cardioids passes through the pole with $a = 1$.

Consider $r = 1 + \text{cos} \theta$,

This polar equation can be written as

$r = 1 \left(1 + \text{cos} \theta \right)$

The above equation is an equation of cardioids passes through the pole with $a = 1$.

To sketch:

$r = 1 + \text{sin} \theta ; r = 1 + \text{cos} \theta$

From the above graph, the point of intersections are $\left(1.207, 1.207\right), \left(0, 0\right) \text{and} \left(- 1.207, - 1.207\right)$.