91Ó°ÊÓ

$r=A(1−0.5\sin \mathrm{θ})$

A collection of quantities is defined as a function of one or more independent variables called parameters in a parametric equation.

Consider the curve $r=\mathrm{A}(1-0.5\sin \mathrm{θ})$

When $A=1$the goal is to draw the cross-section of the curve.

The curve $r=(1-0.5\sin \mathrm{θ})$when $A=1$

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

We obtain the values of the equation $r=(1-0.5\sin \mathrm{θ})$for various $\mathrm{θ}$values.

When $\mathrm{θ}=0$

$$\begin{gathered} r=(1-0.5\sin 0)[\text{since}r=(1-0.5\sin \mathrm{θ}\left)\right] \\ r=(1-0)[\text{since}\sin 0=0] \\ r=1 \end{gathered}$$

Then the coordinate $(r,\mathrm{θ})=(1,0)$

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

$$\begin{gathered} r=\left(1-0.5\sin \frac{\mathrm{π}}{2}\right)[\text{since}r=(1-0.5\sin \mathrm{θ}\left)\right] \\ r=(1-(0.5)·1)\left[\text{since}\sin \frac{\mathrm{π}}{2}=1\right] \\ r=0.5 \end{gathered}$$

Then the coordinate $(r,\mathrm{θ})=\left(0.5,\frac{\mathrm{π}}{2}\right)$

When $\mathrm{θ}=\mathrm{π}$

$$\begin{gathered} r=(1-0.5\sin \mathrm{π})[\text{since}r=(1-0.5\sin \mathrm{θ}\left)\right] \\ r=(1-(0.5)·0)\left[\text{since}\sin \frac{\mathrm{π}}{2}=1\right] \\ r=1 \end{gathered}$$

Then the coordinate $(r,\mathrm{θ})=(1,\mathrm{π})$

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

When $\mathrm{θ}=2\mathrm{π}$

Then the coordinate $(r,\mathrm{θ})=(1,2\mathrm{π})$

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

$(1,0)\left(0.5,\frac{\mathrm{Ï€}}{2}\right)(1,\mathrm{Ï€})\left(1.5,\frac{3\mathrm{Ï€}}{2}\right)(1,2\mathrm{Ï€})$

This is the graphical representation.