The torque TTT (in N⋅\cdot⋅m) exerted on a specialized robotic hinge is modeled as a function of its rotation angle α\alphaα (in radians), where 0<α<2π0 < \alpha < 2\pi0<α<2π.
Show that
cosα(5tanα+2tanα)≡3sinα+2sinα \cos \alpha \left( 5 \tan \alpha + \frac{2}{\tan \alpha} \right) \equiv 3 \sin \alpha + \frac{2}{\sin \alpha} cosα(5tanα+tanα2)≡3sinα+sinα2for α≠nπ2\alpha \neq \frac{n\pi}{2}α=2nπ.
The hinge operates at a specific resistance where the torque is given by the relation T=9sinα−2T = 9 \sin \alpha - 2T=9sinα−2. Hence determine, for 0<α<2π0 < \alpha < 2\pi0<α<2π, the possible values of α\alphaα such that
cosα(5tanα+2tanα)=9sinα−2 \cos \alpha \left( 5 \tan \alpha + \frac{2}{\tan \alpha} \right) = 9 \sin \alpha - 2 cosα(5tanα+tanα2)=9sinα−2giving your answers to 3 significant figures.
