In robotic kinematics, the equilibrium tilt angle α\alphaα of a stabilizer, measured in radians, is modeled by a balance of forces. This equilibrium occurs when the following equation is satisfied:
12sinαcosα2sinα+5=2tanα,sinα≠−2.5\frac{12\sin\alpha \cos\alpha}{2\sin\alpha + 5} = 2\tan\alpha, \quad \sin\alpha \neq -2.52sinα+512sinαcosα=2tanα,sinα=−2.5
Demonstrate that this equilibrium equation can be expressed in the form
6sin3α+2sin2α−sinα=06\sin^3\alpha + 2\sin^2\alpha - \sin\alpha = 06sin3α+2sin2α−sinα=0
Determine the specific tilt angles α\alphaα in the interval −π2<α<π2-\frac{\pi}{2} < \alpha < \frac{\pi}{2}−2π<α<2π that satisfy this condition, giving your answers to 3 decimal places.