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

1 Answer
Aug 19, 2017

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