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.
Show that the function can be simplified as follows:
cos2tsint+sin2tcost≡csct,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
A sensor detects a peak signal intensity when the power P(t)=(cos2tsint+sin2tcost)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)=5−3cottP(t) = 5 - 3\cot tP(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.
Using the result from part (a), or otherwise, find the exact value of the average impulse defined by
∫π6π2(cos2tsint+sin2tcost)cott dt\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \left( \frac{\cos 2t}{\sin t} + \frac{\sin 2t}{\cos t} \right) \cot t \, dt∫6π2π(sintcos2t+costsin2t)cottdt
Practise Edexcel A Level Maths 7.5 Proving Trigonometric Identities with exam-style questions for A Level Maths. 27 questions, matched to the Edexcel A Level Maths (9MA0) 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.