Q. 6 - Calculus

Question

Explain why the graphs of $r=3$ and $r=-3$ are identical in a polar coordinate system.

Step-by-Step Solution

Answer

That means the equations $r=3$ and $r=-3$ are same in polar coordinate system. Therefore $r=3$and $r=-3$ are identical in polar coordinate system. Hence the explanation.

Step 1: Given information

Equations $r=3$ and $r=-3$ are identical in a polar coordinate system

Step 2: calculation

Consider the equations $r=3$ and $r=-3$.

The objective is to show that $r=3$ and $r=-3$ are identical in polar coordinate system.

The value 3 for the polar equation $r=3$ describes the set of points $\left|3\right|$ units from the pole.

Thus $r=3$ is a circle with radius $\left|3\right|$ centered at the pole.

Now for any real number $-3$ the polar equation $r=-3$ describes the set of points $\left|3\right|$ units from the pole.

Thus $r=-3$ is a circle with radius $\left|3\right|$ centered at the pole.

Here 3 represents the length of the initial ray. that means the length of the initial ray is 3 units if $r=3$ or $r=-3$ since the length is positive.

That means the equations $r=3$ and $r=-3$ are same in polar coordinate system. Therefore $r=3$ and $r=-3$ are identical in polar coordinate system.

Hence the explanation.