Q 39. - Calculus

Question

Graph the cardioid $r = 2(1 - \sin \theta )$ For each pair of functions in Exercises $40–45$

Algebraically find all values of $\theta$ where

$f 1\left(\theta \right) = f 2\left(\theta \right)$

Sketch the two curves in the same polar coordinate system.

Find all points of intersection between the two curves.

Step-by-Step Solution

Answer

The graph is 

Step 1: Given information

$r = 2(1 - \sin \theta )$

Step 2: Calculation

Consider the given cardioid, $r=2(1-\sin \theta )$

The goal is to create a graph of the equation.

Assume $\theta =0,\frac{\pi }{6},\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi$

For different $\theta$ values we find the values of the equation $r=2(1-\sin \theta )$

When $\theta =0$

$$\begin{gathered} r=2(1-\sin 0) [\text{ since }r=2(1-\sin \theta \left)\right] \\ r=2(1-0) [\text{ since }\sin 0=0] \\ r=2 \end{gathered}$$

Then the coordinate $(r,\theta )=(2,0)$

When $\theta =\frac{\pi }{6}$ Open with

$$\begin{gathered} r=2\left(1-\sin \frac{\pi }{6}\right)[\text{ since }r=2(1-\sin \theta \left)\right] \\ r=2\left(1-\frac{1}{2}\right)\left[\text{ since }\sin \frac{\pi }{6}=\frac{1}{2}\right] \\ r=1 \end{gathered}$$

Then the coordinate $(r,\theta )=\left(1,\frac{\pi }{6}\right)$

When $\theta =\frac{\pi }{2}$

$$\begin{gathered} r=2\left(1-\sin \frac{\pi }{2}\right)[sincer=2(1-\sin \theta \left)\right] \\ r=2(1-1)\left[\text{ since }\sin \frac{\pi }{2}=1\right] \\ r=0 \end{gathered}$$

Then the coordinate $(r,\theta )=\left(0,\frac{\pi }{2}\right)$

When $\theta =\pi$

$$\begin{gathered} r=2(1-\sin \pi ) [sincer=2(1-\sin \theta \left)\right] \\ r=2(1-0)[\text{ since }\sin \pi =0] \\ r=2 \end{gathered}$$

Then the coordinate $(r,\theta )=(2,\pi )$

When$\theta =\frac{3\pi }{2}$

$$\begin{gathered} r=2\left(1-\sin \frac{3\pi }{2}\right)[sincer=2(1-\sin \theta \left)\right] \\ r=2(1-(-1\left)\right)\left[\text{ since }\sin \frac{3\pi }{2}=-1\right] \\ r=4 \end{gathered}$$

Then the coordinate $(r,\theta )=\left(4,\frac{3\pi }{2}\right)$

When $\theta =2\pi$

$r=2(1-\sin 2\pi ) [$ since $r=2(1-\sin \theta )]$

$r=2(1-0) [$ since $\sin 2\pi =0]$

$r=2$

Then the coordinate $(r,\theta )=(2,2\pi )$

We have distinct coordinates for different $\theta$ values.

To draw the graph, represent all the following points on the graph.

$(2,0)\left(1,\frac{\pi }{6}\right)\left(0,\frac{\pi }{2}\right)(2,\pi )\left(4,\frac{3\pi }{2}\right)(2,2\pi )$

Step 3: Calculation

This is the required graph.