Question

Points `(3 ,2 )` and `(8 ,4 )` are `( pi)/3` radians apart on a circle. What is the shortest arc length between the points?

Answer 1

`s = (pisqrt29)/3`

Explanation:

The square of distance, d, between the two points is:

`d² = (8 - 3)² + (4 - 2)²`

`d² = (5)² + (2)²`

`d² = 29`

d forms one side of a triangle where the other two sides are radii; this allow us the law of cosines to find the radius:

`29 = r² + r² - 2(r)(r)cos(pi/3)`#

`r² = 29/(2 - 2cos(pi/3))`

`r = sqrt29`

The arc length, s, is:

`s = rtheta`

`s = (pisqrt29)/3`