A mechanical sensor measures the angular misalignment $\alpha$ of a rotating component. The misalignment follows the equilibrium condition:

$$6\sin (\alpha -35^{\circ })=-5\cos (\alpha -35^{\circ })$$

Determine all possible values of $\alpha$ in the range $0<\alpha <360^{\circ }$, giving your answers to one decimal place.

Show that the equation describing the lateral vibration of a specific piston,

$$5{\sin }^{3}x=7\sin x-4\sin x\cos x$$

can be written in the form

$$\sin x(a{\cos }^{2}x+b\cos x+c)=0$$

where $a$, $b$ and $c$ are integers to be found.

Hence find all real solutions for $-\pi \le x\le \pi$ to the equation

$$5{\sin }^{3}x=7\sin x-4\sin x\cos x$$

giving your answers to two decimal places where appropriate.