Trigonometric Moments of a Circular Data Set

Hi,

Is there a simple relationship between the angles obtained from the
first and second moments for a circular data set?

That is, given a collection of N points distributed on a unit circle
(with angles q1,...,qN), if Theta1 is the angle from the first trig
moment, i.e. the angle that satisfies:
(i) Theta1 = ArcTan[ Sum_i=1,N Sin(qi) / Sum_i=1,N Cos(qi)]
and Theta2 is the angle from the second trig moment:
(ii) Theta2 = ArcTan[ Sum_i=1,N Sin(2*qi) / Sum_i=1,N Cos(2*qi)]

Is there a direct, analytical relationship between Theta1 and Theta2?

TIA,

Juno

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