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.
317 exam-style questions on OCR A Level Maths 1.5 Trigonometry, covering 1.5.1 Definitions for all arguments, 1.5.2 Sine and cosine rules, 1.5.3 Area of a triangle, 1.5.4 Radian measure (A-level only), 1.5.5 Small angle approximations (A-level only), 1.5.6 Graphs of basic trigonometric functions, 1.5.7 Exact values in radians (A-level only), 1.5.8 Reciprocal and inverse trigonometric ratios (A-level only), 1.5.9 Graphs of reciprocal and inverse functions (A-level only), 1.5.10 Trigonometric identities, 1.5.11 Further trigonometric identities (A-level only), 1.5.12 Double angle and compound angle formulae (A-level only), 1.5.13 Geometrical proofs of formulae (A-level only), 1.5.14 Harmonic form Rcos / Rsin (A-level only), 1.5.15 Trigonometric equations, 1.5.16 Proof involving trigonometric functions (A-level only), and 1.5.17 Trigonometric functions in context. Each one has a worked solution and a mark scheme showing where the marks go.