A robotic arm's vertical extension EEE (in cm) during a cycle is given by the function
E(t)=20sintcost−16sin2t+11,t≥0 E(t) = 20 \sin t \cos t - 16 \sin^2 t + 11, \quad t \ge 0 E(t)=20sintcost−16sin2t+11,t≥0where t t\,t is the time in seconds.
Write E(t)E(t)E(t) in the form asin2t+bcos2t+ca \sin 2t + b \cos 2t + casin2t+bcos2t+c, where a,b, a, b,\,a,b, and c c\,c are integers to be determined.
Use your result from part (a) to express E(t)E(t)E(t) in the form
Rsin(2t+α)+c R \sin(2t + \alpha) + c Rsin(2t+α)+cwhere 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 extension of the robotic arm, (ii) find the second smallest positive value of t t\,t at which this maximum extension occurs, giving your answer to 3 significant figures.
