Question

Are the two polar coordinates the same: `(1, 93pi/4), (-1, pi/4)`?

Answer 1

These polar coordinates are not the same. One corresponds to an upward vertical line, 1 unit in length, and the other to a downward vertical line of the same length.

Explanation:

Polar coordinates are in the form `(r, theta)` where `r` is the distance from the origin and `theta` is the angle counterclockwise from the positive `x` axis.

It's possible... their distances from the origin are the same. `pi/2=180^o` so `pi/2=90` counterclockwise from the positive `x` axis, means that the second-named point is an upward vertical line -1 units long, which in practice is downward vertical line 1 unit long.

`2 pi` is a full rotation that brings us back to the starting point, so `92 pi` is just 46 full rotations, and we can delete it. The polar coordinates of the first-named point, then, are equivalent to `(1, pi/4)`. As described above, this is an upward vertical line, 1 unit in length.

Since we discovered that the other point is a downward vertical line one unit in length, we can confidently state that these polar coordinates are not the same.