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1.5 Trigonometry

1.5 Trigonometry

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Question 98

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.

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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]
Markscheme

1.5 Trigonometry Questions

  1. A Level
  2. /Maths
  3. /1.5 Trigonometry

317 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.

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