Question

What is the distance between the following polar coordinates?: `(8,(-21pi)/12), (5,(-3pi)/8)`

Answer 1

`10.937`

Explanation:

First `(8,-(21pi)/12)` can be simplified a bit `(8, -(7pi)/4)`, and since `-(7pi)/4` is coterminal to `pi/4`, we'll use `(8,pi/4)` as an equivalent point. The other point we'll keep as `(5,-(3pi)/8)`.

If we plot the points and use them as two vertices of a triangle with the origin as the third vertex, we have sides 8 and 5 with an angle of `pi/4 + (3pi)/8 = (5pi)/8` between them.

For this triangle we can use the Law of Cosines to find the side opposite `(5pi)/8`, which is the distance between the given points:

`c=sqrt(a^2+b^2-2abcos(C))`

`c=sqrt(8^2+5^2-2(8)(5)cos((5pi)/8))`

`c approx 10.937`

This video might also be helpful.