The power output PPP, in kilowatts, of a rotational component is modeled by the function
P(θ)=14sinθcosθ+6cos2θ−5 P(\theta) = 14 \sin \theta \cos \theta + 6 \cos^2 \theta - 5 P(θ)=14sinθcosθ+6cos2θ−5where θ \theta\,θ is the angular displacement in radians for 0≤θ<2π0 \le \theta < 2\pi0≤θ<2π.
Write P(θ)P(\theta)P(θ) in the form asin2θ+bcos2θ+ca \sin 2\theta + b \cos 2\theta + casin2θ+bcos2θ+c, where a,b, a, b,\,a,b, and c c\,c are integers to be found.
Using your result from part (a), express P(θ)P(\theta)P(θ) in the form Rsin(2θ+α)+cR \sin (2\theta + \alpha) + cRsin(2θ+α)+c where R>0 R > 0\,R>0 and 0<α<π2\displaystyle 0 < \alpha < \frac{\pi}{2}0<α<2π. Give the exact value of R R\,R and the value of α \alpha\,α in radians to 3 significant figures.
Hence, or otherwise, (i) state the maximum value of P(θ)P(\theta)P(θ) predicted by this model, (ii) find the second smallest positive value of θ \theta\,θ at which this maximum power output occurs. Give your answer to 3 significant figures.
