Question

What is the distance between the following polar coordinates?: `(1,(-23pi)/12), (3,(5pi)/8)`

Answer 1

The distance, `c ~~ 3.28`

Explanation:

When given two polar points `(r_1,theta_1)` and `(r_2,theta_2)`, please observe that the two radii `r_1` and `r_2` form a triangle with the line between the two points. Here is a graph of the, two points, the two radii, and the line between them:

We can use the Law of Cosines to find the length, c, of the blue line:

`c^2 = r_1^2 + r_2^2 -2(r_1)(r_2)cos(theta_2-theta_1)" [1]"`

Using the two given points we assign:

`r_1 = 1`, `r_2=3`, `theta_1 = -(23pi)/12` and `theta_2 = (5pi)/8`

Substituting into equation [1]:

`c^2 = 1^2 + 3^2 -2(1)(3)cos((5pi)/8-(-(23pi)/12))`

`c ~~ 3.28`