Length of the chord is given by L = 2 R Sin Ф,
where R is the radius of the circle and Ф is the angle subtended by the chord at the center of the circle.
If chord length is doubled then, Sin Ф' / Sin Ф = L' / L
=> Sin Ф' / Sin Ф = 2
=> Ф' = Sin⁻¹ [ 2 Sin Ф ] Ф' <= π/3 and Ф <= π/2
for very small angles Ф, the angle also nearly doubles. Like for 15 degs or less.