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.
317 exam-style questions on OCR (MEI) A Level Maths 1.7 Trigonometry, covering 1.7.1 Solve right-angled triangles, 1.7.2 Definitions of sin, cos and tan for any angle, 1.7.3 Graphs of sin, cos and tan, 1.7.4 Exact values of trig functions (degrees), 1.7.5 Area of a triangle, 1.7.6 Sine and cosine rules, 1.7.7 Identity tan = sin/cos, 1.7.8 Identity sin^2 + cos^2 = 1, 1.7.9 Solve simple trigonometric equations, 1.7.10 Exact values of trig functions (radians) (A-level only), 1.7.11 Inverse trigonometric functions (A-level only), 1.7.12 Radians and degree conversion (A-level only), 1.7.13 Arc length and area of a sector (A-level only), 1.7.14 Small angle approximations (A-level only), 1.7.15 Sec, cosec and cot functions (A-level only), 1.7.16 Graphs of reciprocal trig functions (A-level only), 1.7.17 Pythagorean identities for sec and cosec (A-level only), 1.7.18 Compound angle formulae (A-level only), 1.7.19 Double angle identities (A-level only), 1.7.20 Expressions for a cos θ ± b sin θ (A-level only), 1.7.21 Use identities to solve equations (A-level only), 1.7.22 Proofs involving trigonometric functions (A-level only), 1.7.23 Trig to solve problems in context (A-level only), and 1.7 Trigonometry. Each one has a worked solution and a mark scheme showing where the marks go.