Points `(4 ,4 )` and `(7 ,3 )` are `(5 pi)/4` radians apart on a circle. What is the shortest arc length between the points?

Because the angle `(5pi)/4` is greater than `pi`, we know that this is NOT the smallest angle between the two points; the smallest angle is:

`angle theta=2pi-(5pi)/4`

`angle theta = (3pi)/4`

This will be the angle between the two radii that connect the two points `(4,4)` and `(7,3)`. The two radii and the chord, c, between the two points form a triangle, therefore, we can write an equation, using the Law of Cosines:

`c^2 = r^2 + r^2-2(r)(r)cos(theta)`

We know that `c^2` is the square of the distance between the two points and we know the value of `theta`:

`(7-4)^2+(3-4)^2 = r^2 + r^2-2(r)(r)cos((3pi)/4)`

Remove a common factor of `r^2`:

`3^2+(-1)^2= r^2(2-2cos((3pi)/4))`

Evaluate the cosine function:

`10= r^2(2-2(-sqrt2/2))`

Simplify:

`10= r^2(2+sqrt2)`

Multiply both sides by the conjugate:

`10(2-sqrt2)= r^2(2+sqrt2)(2-sqrt2)`

`10(2-sqrt2)= 6r^2`

Flip the equation and divide both sides by 6:

`r^2 = 5/3(2-sqrt2)`

Use the square root operation on both sides:

`r = sqrt(5/3(2-sqrt2))`

We know that the arclength, S, is the radius multiplied by the radian measure of angle:

`S = thetar`

`S = (3pi)/4sqrt(5/3(2-sqrt2))`