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

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

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.