Question

What is the distance between `(3 , (3 pi)/8 )` and `(9, pi )`?

Answer 1

`sqrt(90-54cos(5pi/8))` = `sqrt110.665=10.52` nearly.

Explanation:

The position vectors to the points are of lengths a = 3 and b = 9. The angle in-between is C = `5pi/8`.
Use the formula c = `sqrt(a^2+b^2-2ab cos C)`

Answer 2

`sqrt{90 + 54cos({3pi}/8)} ~~ 10.520`

Explanation:

To convert the polar coordinates to Cartesian coordinates, we use

`x = r cos(theta)`
`y = r sin(theta)`

The cartesian coordinate of `(3,{3pi}/8)` is `(3/2sqrt(2-sqrt2),3/2sqrt(2+sqrt2))`. Use the half angle formula to get the values.

The cartesian coordinate of `(9,pi)` is `(-9,0)`.

We can use the Pythagoras Theorem to find the distance between the 2 points

`d = sqrt{(-9 - 3cos({3pi}/8))^2 + (0 - 3sin({3pi}/8))^2}`

`= sqrt{(81 + 9cos^2({3pi}/8) + 54cos({3pi}/8)) + 9sin^2({3pi}/8)}`

`= sqrt{90 + 54cos({3pi}/8)}`

`= 3sqrt{3sqrt{2-sqrt2}+10}`

`~~ 10.520`