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 -2.5 2sinα+512sinαcosα=2tanα,sinα=−2.5Demonstrate that this equilibrium equation can be expressed in the form
6sin3α+2sin2α−sinα=0 6\sin^3\alpha + 2\sin^2\alpha - \sin\alpha = 0 6sin3α+2sin2α−sinα=0Determine 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 AQA A Level Maths 1.8 E: Trigonometry with exam-style questions for A Level Maths. 239 questions covering 1.8.1 Trigonometric definitions, rules and radians, 1.8.2 Small angle approximations (A-level only), 1.8.3 Trigonometric functions and exact values, 1.8.4 Reciprocal and inverse trigonometric functions (A-level only), 1.8.5 Trigonometric identities, 1.8.6 Compound and double angle formulae (A-level only), 1.8.7 Solving trigonometric equations, 1.8.8 Proofs with trigonometric identities, and 1.8.9 Trigonometry in context, matched to the AQA A Level Maths (7357) 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.
