91Ó°ÊÓ

$(r,\mathrm{θ})$

A polar curve is a shape constructed using the polar coordinate system.

Consider the polar coordinate $(r,\mathrm{θ})$

The goal is to show that for any $k$$(r,theta+2pik)$denotes the same point.

The point $(r,theta+2pik)$completes the revolutions and reaches the same position for any value of the integer $k$

That is, if $k=1$one revolution is completed. It completes two revolutions when $k=2$

Thus for any point $(r,\mathrm{θ})$the point $(r,\mathrm{θ}+2\mathrm{π}k)$gives the same point since we are adding $2\mathrm{π}$multiple.

In the graph, we can observe that

Example: consider the point $\left(2,\frac{\mathrm{Ï€}}{6}\right)$

$$\begin{gathered} \left(2,\frac{\mathrm{Ï€}}{6}\right)=\left(2,\frac{\mathrm{Ï€}}{6}+2\mathrm{Ï€}\right) \\ =\left(2,\frac{13\mathrm{Ï€}}{6}\right) \end{gathered}$$

The representation of the point $\left(2,\frac{\mathrm{Ï€}}{6}\right)$is same as $\left(2,\frac{13\mathrm{Ï€}}{6}\right)$

This is the explanation.

Consider the point $(r,\mathrm{θ})$

Objective is to prove that $(-r,\mathrm{θ}+\mathrm{π})$is same as $(r,\mathrm{θ})$

If we add an angle $\mathrm{π}$to an angle $\mathrm{θ}$in the polar coordinate system, we get the same angle.

Example: consider the polar coordinate $\left(2,\frac{\mathrm{Ï€}}{6}\right)$

$$\begin{gathered} \left(2,\frac{\mathrm{Ï€}}{6}\right)=\left(-2,\frac{\mathrm{Ï€}}{6}+\mathrm{Ï€}\right) \\ =\left(-2,\frac{7\mathrm{Ï€}}{6}\right) \end{gathered}$$

Therefore the point $\left(2,\frac{\mathrm{Ï€}}{6}\right)$and $\left(-2,\frac{7\mathrm{Ï€}}{6}\right)$represents the same point. This is the explanation.

Consider the point $(0,\mathrm{θ})$

Objective is to prove that $(0,\mathrm{θ})$is the pole for any value of $\mathrm{θ}$

From the graph we can observe that for any value of $\mathrm{θ}$let the points $(0,\mathrm{θ})\left(0,\frac{\mathrm{π}}{3}\right)$represent the pole.

This is the explanation.