91Ó°ÊÓ

$r=3+\cos \mathrm{θ},r=3+3\cos \mathrm{θ},r=3+4\cos \mathrm{θ},r=3+6\cos \mathrm{θ}$

Consider the limacon $r=3+\cos \mathrm{θ}$

The goal is to figure out how graphs of the pattern $r=3+b\cos \mathrm{θ}$behave.

Find different $r$values by substituting different $\mathrm{θ}$values.

Consider $\mathrm{θ}=0,\frac{\mathrm{π}}{2},\mathrm{π},\frac{3\mathrm{π}}{2},2\mathrm{π}$

For different $\mathrm{θ}$values, we find the values of the equation $r=3+\cos \mathrm{θ}$

When $\mathrm{θ}=0$

$$\begin{gathered} r=3+\cos 0[\text{since}r=3+\cos \mathrm{θ}] \\ r=3+1\text{since}\cos 0=1] \\ r=4 \end{gathered}$$

Then the coordinate $(r,\mathrm{θ})=(4,0)$

When $\mathrm{θ}=\frac{\mathrm{π}}{2}$

$$\begin{gathered} r=3+\cos \frac{\mathrm{π}}{2}[\text{since}r=3+\cos \mathrm{θ}] \\ r=3+0\left[\text{since}\cos \frac{\mathrm{π}}{2}=0\right] \\ r=3 \end{gathered}$$

Then the coordinate $(r,\mathrm{θ})=\left(3,\frac{\mathrm{π}}{2}\right)$

When $\mathrm{θ}=\mathrm{π}$

$$\begin{gathered} r=3+\cos \mathrm{π}[\text{since}r=3+\cos \mathrm{θ}] \\ r=3+(-1)\left[\text{since}\cos \frac{\mathrm{π}}{2}=0\right] \\ r=2 \end{gathered}$$

Then the coordinate $(r,\mathrm{θ})=(2,\mathrm{π})$

When $\mathrm{θ}=\frac{3\mathrm{π}}{2}$

Open with -

$$\begin{gathered} r=3+\cos \frac{3\mathrm{π}}{2}[\text{since}r=3+\cos \mathrm{θ}] \\ r=3+0\left[\text{since}\cos \frac{3\mathrm{π}}{2}=0\right] \\ r=3 \end{gathered}$$

When $\mathrm{θ}=2\mathrm{π}$

$$\begin{gathered} r=3+\cos 2\mathrm{π}[\text{since}r=3+\cos \mathrm{θ}] \\ r=3+1[\text{since}\cos 2\mathrm{π}=1] \\ r=4 \end{gathered}$$

Then the coordinates are $(r,\mathrm{θ})=(4,2\mathrm{π})$

We have distinct coordinates for different $\mathrm{θ}$values.

Draw the graph by putting all of the following points on it.

Similarly, we can find the values of $r$$r\cos r=3+4\cos \mathrm{θ}$and $r=3+6\cos \mathrm{θ}$

Now plot the points on the graph and draw the different graphs.

The inner loop grows in size as the $b$value increases in the graph. The equation $r=3+\cos \mathrm{θ}$is a cardioid. when $0≤b≤\frac{3}{2}$the graph is convex. When $\frac{3}{2}<b≤3$the graph is a dimple.

Hence this is the explanation.