A mechanical sensor measures the angular misalignment α \alpha\,α of a rotating component. The misalignment follows the equilibrium condition: 6sin(α−35∘)=−5cos(α−35∘)6 \sin(\alpha - 35^\circ) = -5 \cos(\alpha - 35^\circ)6sin(α−35∘)=−5cos(α−35∘) Determine all possible values of α \alpha\,α in the range 0<α<360∘0 < \alpha < 360^\circ0<α<360∘, giving your answers to one decimal place.
Show that the equation describing the lateral vibration of a specific piston, 5sin3x=7sinx−4sinxcosx5 \sin^3 x = 7 \sin x - 4 \sin x \cos x5sin3x=7sinx−4sinxcosx can be written in the form sinx(acos2x+bcosx+c)=0\sin x (a \cos^2 x + b \cos x + c) = 0sinx(acos2x+bcosx+c)=0 where aaa, b b\,b and c c\,c are integers to be found.
Hence find all real solutions for −π≤x≤π-\pi \le x \le \pi−π≤x≤π to the equation 5sin3x=7sinx−4sinxcosx5 \sin^3 x = 7 \sin x - 4 \sin x \cos x5sin3x=7sinx−4sinxcosx giving your answers to two decimal places where appropriate.