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3 Unit A2 1: Pure Mathematics (A-level only)

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Question 105

In a study of signal interference, the intensity of a resultant wave is modeled by a function containing trigonometric ratios.

a.

Prove that

cos⁡2ϕsin⁡ϕ+sin⁡2ϕcos⁡ϕ≡csc⁡ϕ,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∈Z
[3]
b.

Hence solve, for 0≤θ<2π0 \le \theta < 2\pi0≤θ<2π,

2(cos⁡4θsin⁡2θ+sin⁡4θcos⁡2θ)+3cot⁡22θ=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=5

giving your answers in radians to 3 significant figures where appropriate.

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3 Unit A2 1: Pure Mathematics (A-level only) Questions

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