In a study of signal interference, the intensity of a resultant wave is modeled by a function containing trigonometric ratios.
Prove that
cos2ϕsinϕ+sin2ϕcosϕ≡cscϕ,ϕ≠nπ2,n∈Z \frac{\cos 2\phi}{\sin \phi} + \frac{\sin 2\phi}{\cos \phi} \equiv \csc \phi, \quad \phi \neq \frac{n\pi}{2}, n \in \mathbb{Z} sinϕcos2ϕ+cosϕsin2ϕ≡cscϕ,ϕ=2nπ,n∈ZHence solve, for 0≤θ<2π0 \le \theta < 2\pi0≤θ<2π,
2(cos4θsin2θ+sin4θcos2θ)+3cot22θ=5 2 \left( \frac{\cos 4\theta}{\sin 2\theta} + \frac{\sin 4\theta}{\cos 2\theta} \right) + 3\cot^2 2\theta = 5 2(sin2θcos4θ+cos2θsin4θ)+3cot22θ=5giving your answers in radians to 3 significant figures where appropriate.
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.
