Question

if r does not equal 0, which of the following polar coordinate pairs represents the same point as the point with polar coordinates (r,θ)? ( 2 answers required only)

Answer 1

In summary, the polar coordinate pairs that represent the same point as
`$(r, \theta)$` when
`$r \\eq 0$` are:

1.
`$(r, \theta-2 \pi)$`

2.
`$(r, \theta-3 \pi)$`

To determine which of the given polar coordinate pairs represents the same point as the point with polar coordinates
`$(r, \theta)$` when
`$r \\eq 0$`, we can use the following relationships in polar coordinates:

1.
`$(r, \theta)$` represents a point with distance
`$r$` from the origin and an angle of
`$\theta$` from the positive x-axis.

Now, let's examine each of the options one by one:

1.
`$(r, \theta-2 \pi)$`

This represents a point with the same distance
`$r$` from the origin, but the angle is
`$\theta - 2\pi$`, which is the same as
`$\theta$` since subtracting
`$2\pi$` from an angle is the same as not changing the angle at all.

So, this coordinate pair represents the same point:
`$(r, \theta-2 \pi) = (r, \theta)$`.

2.
`$(r, \theta+\pi)$`

This represents a point with the same distance
`$r$` from the origin, but the angle is
`$\theta + \pi$`. Adding
`$\pi$` to an angle means rotating the point by
`$\pi$` radians in the opposite direction (180 degrees).

So, this coordinate pair represents the same point:
`$(r, \theta+\pi)$` is the opposite direction but the same distance, so it's equivalent to
`$(-r, \theta)$`.

3.
`$(-r, \theta-\pi)$`

This represents a point with the distance
`$-r$`, which doesn't make sense in polar coordinates because distances should be positive. So, this option is not valid.

4.
`$(r, \theta-3 \pi)$`

This represents a point with the same distance
`$r$` from the origin, but the angle is
`$\theta - 3\pi$`. Subtracting
`$3\pi$` from an angle is the same as subtracting
`$2\pi$` (a full rotation) plus an additional
`$\pi$` radians, which is equivalent to rotating by
`$\pi$` radians in the opposite direction.

So, this coordinate pair represents the same point:
`$(r, \theta-3 \pi)$` is equivalent to
`$(-r, \theta)$`.

`$(r, \theta-3 \pi)$`

Option 3,
`$(-r, \theta-\pi)$`, is not valid as it has a negative distance.

Answer 2

First option

(r, θ - 2π)

Explanation:

Polar coordinates are a representation in terms of a given longitude r, and an orientation given by an angle. (Please see attached image)

Since the polar coordinate is cyclic (The values from the angle go from 0 to 2π and then go through the same values again) if we add or subtract 2π, from any value we find ourselves in the same initial position

So, any point can be represented by

(r, θ ± 2π)