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∈ZA 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)=5−3cott P(t) = 5 - 3\cot t P(t)=5−3cottHence 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
