Question

What is the distance between `(-6 , pi/2 )` and `(5, pi/6 )`?

Answer 1

`\sqrt{91}` units.

Explanation:

Since the coordinates are polar coordinates, we imagine a triangle `OAB` with `O` at the origin, `A` at `(-6,\pi/2)` and `B` at `(5,\pi/6)`. We will use the Law of Cosines to get the length of `AB`.

First convert side `OA` to a positive length by writing its coordinates as `(+6,-\pi/2)`, changing the sign of the radius and compensating by subtracting `\pi` from the angle.

So `OA=6` and `OB=5`. Next we need angle `O` which is the difference between the angular coordinates after we have made the radial coordinates positive (see above). Thus

`\pi/2-(-\pi/6)=2\pi/3`.

Now apply the Law of Cosines:

`(AB)^2=(OA)^2+(OB)^2-2(OA)(OB)\cos(angle O)`

`=6^2+5^2-2(6)(5) \cos(2\pi/3)`

As `\cos(2\pi/3)=-1/2` this gives `(AB)^2=91`.