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1.8 E: Trigonometry

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Question 120

A signal processing engineer models a modulated acoustic wave using the function S(t)S(t)S(t) for 0<t<π0 < t < \pi0<t<π, where t t\,t is the time in milliseconds.

a.

Show that the function can be simplified as follows:

cos⁡2tsin⁡t+sin⁡2tcos⁡t≡csc⁡t,t≠nπ2,  n∈Z \frac{\cos 2t}{\sin t} + \frac{\sin 2t}{\cos t} \equiv \csc t, \quad t \neq \frac{n\pi}{2}, \; n \in \mathbb{Z} sintcos2t​+costsin2t​≡csct,t=2nπ​,n∈Z
[3]
b.

A sensor detects a peak signal intensity when the power P(t)=(cos⁡2tsin⁡t+sin⁡2tcos⁡t)2\displaystyle P(t) = \left( \frac{\cos 2t}{\sin t} + \frac{\sin 2t}{\cos t} \right)^2P(t)=(sintcos2t​+costsin2t​)2 satisfies the condition

P(t)=5−3cot⁡t P(t) = 5 - 3\cot t P(t)=5−3cott

Hence solve, for 0<t<π0 < t < \pi0<t<π, the equation above, giving your answers to 3 significant figures as appropriate.

[5]
c.

Using the result from part (a), or otherwise, find the exact value of the average impulse defined by

∫π6π2(cos⁡2tsin⁡t+sin⁡2tcos⁡t)cot⁡t dt \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \left( \frac{\cos 2t}{\sin t} + \frac{\sin 2t}{\cos t} \right) \cot t \, dt ∫6π​2π​​(sintcos2t​+costsin2t​)cottdt
[3]

1.8 E: Trigonometry Questions

  1. A Level
  2. /Maths
  3. /1.8 E: Trigonometry

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.

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