Trap e zium

Algebra Level 5

For how many real values of the parameter \(a\) is the following true?

If for every solution \(\theta\) of the equation:

\(6\sin(2\theta+\frac{11\pi a}{6}) - 6\sin (\frac{11\pi a}{6})-24a^3-28a^2-6a+1\)
\(=2(12a^2+8a-1)\cos(\theta+\frac{11\pi a}{6}) - 6(2a+1)\sin \theta,\)

we draw a point \((\cos \theta, \sin \theta)\), then we will obtain exactly four points that form a trapezium.

Details and assumptions

A trapezium has a pair of parallel sides


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