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.
Practise Edexcel A Level Maths Trigonometry and Modelling with exam-style questions for A Level Maths. 100 questions covering 7.1 Addition Formulae, 7.2 Double Angle Formulae, 7.3 Solving Trigonometric Equations, 7.4 Simplifying a cos x +- b sin x, 7.5 Proving Trigonometric Identities, and 7.6 Modelling with Trigonometric Functions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.