Q. 8 - Calculus

Question

Explain why the graphs of $\theta =\frac{\pi }{2}$ and $\theta =-\frac{\pi }{2}$ are identical in a polar coordinate system.

Step-by-Step Solution

Answer

The graphical representation of $\theta =\frac{\pi }{2}$ and $\theta =-\frac{\pi }{2}$

The equations $\theta =\frac{\pi }{2}$ and $\theta =-\frac{\pi }{2}$ are identical in polar coordinate system.

Step 1: Given information

The equations $\theta =\frac{\pi }{2}$ and $\theta =-\frac{\pi }{2}$

Step 2: Calculation

Consider the equations $\theta =\frac{\pi }{2}$ and $\theta =-\frac{\pi }{2}$.

The objective is to show that $\theta =\frac{\pi }{2}$ and $\theta =-\frac{\pi }{2}$ are same in polar coordinate system.

For the value of $\theta$, the polar equation $\theta =\frac{\pi }{2}$ describes the set of points with polar angle $\frac{\pi }{2}$.

As the angle is positive it moves in the counterclockwise direction.

When $\theta =\frac{\pi }{2}$ the value of $r$ can be either positive or negative.

Thus, the graph for $\theta =\frac{\pi }{2}$ is line through the pole.

When $\theta =-\frac{\pi }{2}$ the value of $r$ can be either positive or negative. and $\theta =-\frac{\pi }{2}$ lies on the same line that of $\theta =\frac{\pi }{2}$ which passes through the pole.

As the angle is negative it moves in the clockwise direction.

The graphical representation of $\theta =\frac{\pi }{2}$ and $\theta =-\frac{\pi }{2}$

That means the equations $\theta =\frac{\pi }{2}$ and $\theta =-\frac{\pi }{2}$ are identical in polar coordinate system. Hence the explanation.