Calculation using complex numbers and vectors

Let and . The first segment can then be represented by the vector . Representing the other segments as vectors, we can add the vectors to get a sum that must equal 1, that is

This calculation is easy to do numerically to give

This shows that r = 0.34845 and A = 11.8186°.

To compute the exact value of r, we must be more careful with the calculation. Using DeMoivre's Theorem, we first get that

Now taking the reciprical yields

The absolute value of this complex number shows us that

There is really no advantage in trying to find an exact value for the angle A. But.....if you really need to know, see the derivation using geometry and trigonometry.