Question

Which of the following is NOT the same point in polar coordinates as (3, -1.236)? (-3, 1.906) (3, -7.518) (3, 5.047) (-3, 1.236)

Answer 1

`(-3, 1.236)`

Explanation:

Given:

`(3, -1.236)`

`(-3, 1.906)`

`(3, -7.518)`

`(3, 5.047)`

`(-3, 1.236)`

Note that:

`7.518-1.236 = 6.282 ~~ 2pi`

`5.047+1.236 = 6.283 ~~ 2pi`

`1.236+1.906 = 3.142 ~~ pi`

If the sign of `r` is inverted, then to get the same point you need the signs of `cos theta` and `sin theta` to both be inverted. This can be done by replacing `theta` with `theta+pi`

`sin` and `cos` both have period `2pi`.

So we can tell:

• The first and second coordinates are equivalent, since `1.906 ~~ -1.236+pi` and the sign of `r` is inverted.

• The first and third coordinates are equivalent, since the `theta` values differ by `2pi`.

• The first and fourth coordinates are equivalent, since the `theta` values differ by `2pi`.

• The fifth coordinates are not equivalent: Inverting the signs of `r` and `theta` does not generally give the same point.