Question

What is the distance between the following polar coordinates?: `(7,(-2pi)/3), (5,(-pi)/6)`

Answer 1

`sqrt74` units.

Explanation:

It's probably easiest to do this by first converting your polar coordinates into Cartesian coordinates—of course, polar coordinates are non-unique so theoretically they could translate to different Cartesian coordinates. We shall take the obvious set of points, however.

To convert, remember that for a set of polar coordinates `[r, theta]`, `r=sqrt(x^2+y^2)`. Secondly, `x=r cos theta` and `y= r sin theta`.

For the point `[7, -(2pi)/3]`, we get:
`x=7cos(-(2pi)/3)` and `y=7sin(-(2pi)/3)`.

`cos(-(2pi)/3) = cos((2pi)/3) = -cos(pi/3) = -1/2`
`sin(-(2pi)/3) = -sin((2pi)/3) = -sin(pi/3) = -sqrt3/2`

With a bit of further calculation:

`x=-7/2` and `y=(-7sqrt3)/2`

For the point `[5, -pi/6]`, we get:
`x=5cos(-pi/6)` and `y=5sin(-pi/6)`.

`cos(-pi/6) = cos(pi/6) = sqrt3/2`.
`sin(-pi/6) = -sin(pi/6) = 1/2`.

With a further bit of calculation:

`x=(5sqrt3)/2` and `y=-5/2`.

The next bit is the easiest; simply apply the formula for the distance between two points `D_"xy"=sqrt((x_2-x_1)^2+(y_2-y_1)^2)` with the two points `(-7/2,(-7sqrt3)/2)` and `((5sqrt3)/2, -5/2)`.

The final answer after inserting those numbers is:

`D_"xy"=sqrt74`