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1.7 Trigonometry

1.7 Trigonometry

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Question 95

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]
Markscheme

1.7 Trigonometry Questions

  1. A Level
  2. /Maths
  3. /1.7 Trigonometry

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.

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