Trap e zium

Algebra Level 5

For how many real values of the parameter aa is the following true?

If for every solution θ\theta of the equation:

6sin(2θ+11πa6)6sin(11πa6)24a328a26a+16\sin(2\theta+\frac{11\pi a}{6}) - 6\sin (\frac{11\pi a}{6})-24a^3-28a^2-6a+1
=2(12a2+8a1)cos(θ+11πa6)6(2a+1)sinθ,=2(12a^2+8a-1)\cos(\theta+\frac{11\pi a}{6}) - 6(2a+1)\sin \theta,

we draw a point (cosθ,sinθ)(\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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