In a stability analysis for a rotating pendulum, the equilibrium angle α\alphaα in the interval 0<α≤π0 < \alpha \le \pi0<α≤π satisfies the equation
2cosec2α−5cotα=5 2\text{cosec}^2 \alpha - 5\cot \alpha = 5 2cosec2α−5cotα=5Determine all possible values for α\alphaα, giving your answers to 3 significant figures.
Prove the following trigonometric identity for all valid values of θ\thetaθ:
sin8θsin3θ−cos8θcos3θ≡sin5θsin3θcos3θ \frac{\sin 8\theta}{\sin 3\theta} - \frac{\cos 8\theta}{\cos 3\theta} \equiv \frac{\sin 5\theta}{\sin 3\theta \cos 3\theta} sin3θsin8θ−cos3θcos8θ≡sin3θcos3θsin5θ317 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.