problem is below in information

Question

Find the lengths of both circular arcs on the unit circle connecting the point and the point that makes an angle of with the positive horizontal axis.
Enter your answer as a comma-separated list, with no decimal approximations.

Answer 1

The point (√3/2,1/2) creates the radius which is inclined at an angle of 30° to the positive x-axis, because tan(30)=(1/2)/(√3/2)=1/√3. The difference between 130° and 30°=100°, subtended by a small arc and large arc, the sum of the lengths of which make up the circumference of the unit circle (radius 1).

The ratio of the length of the small arc (s) to the circumference (=2πr=2π) is also the ratio of the angle 100° to 360°=100/360. The length of the large arc (S) is 2π-(small arc length).

s/(2π)=100/360, so s=200π/360=5π/9, and S=2π-5π/9=13π/9.

SOLUTION: 5π/9, 13π/9