1. Find the lengths of both circular arcs of the unit circle connecting the point PreCalc?

Refer to the figure below.
A is the endpoint of the radius that makes an angle of 20° with the positive horizontal.
O (0,0) is the center of the circle.
Radius of the unit circle = 1

tan(θ) = |(-1/2)/(√3/2)
tan(θ) = 1/√3
θ = 30°
Hence, ∠AOB = 20° + 30°
∠AOB = 50°

Minor arc AB
= Circumference × (50/360)
= 2 × π × 1 × (50/360)
= 5π/18 (units)

Major arc AB
= Circumference × [(360 - 50)/360]
= 2 × π × 1 × (310/360)
= 31π/18 (units)

Find the lengths of both circular arcs of the unit circle 
connecting the point (−√3/2, −1/2) and the endpoint of 
the radius that makes an angle of 20° with the positive horizontal axis.
Radius of the unit circle = 1
tan (θ) = (-1/2)/(√3/2)
tan (θ) = 1/√3
θ = -30°
Hence, ∠AOB = 20° - 30°

∠AOB = -10°

Minor arc AB

= Circumference × (50/360) 
= 2 × π × 1 × (50/360) 
= 5π/18 (units) 
Major arc AB 
= Circumference × [(360 - 50)/360] 
= 2 × π × 1 × (310/360) 
= 31π/18 (units)