Question

What is the distance between the following polar coordinates?: `(3,(23pi)/12), (7,(13pi)/8)`

Answer 1

The distance between the two points is `5.7`

Explanation:

It's easier to calculate with cartesian coordinates than with polar, so my advice is to start with converting `(3;(23pi)/12)` and `(7;(13pi)/8)` to `(x_1;y_1)` and `(x_2;y_2)`

The relationship between `color(blue)(cartesian)` and `color(red)(polar)` coordinates is:

`color(blue)(x)=color(red)(R cdot cos(theta)`
`color(blue)(y)=color(red)(R cdot sin(theta)`

For your first point:
`R=3`
`theta=(23pi)/12`

For your second point:
`R=7`
`theta=(13pi)/8`

The distance between two points is:
`d=sqrt((x_1-x_2)^2+(y_1-y_2)^2)`

`=sqrt((3cos((23pi)/12)-7cos((13pi)/8))^2+(3sin((23pi)/12)-7sin((13pi)/8))^2)`

`=sqrt((3cos(-pi/12)-7cos(-(3pi)/8))^2+(3sin(-pi/12)-7sin(-(3pi)/8))^2)`

`=sqrt((3cos(pi/12)-7cos((3pi)/8))^2+(7sin((3pi)/8)-3sin(pi/12))^2)`

`~~5.7`