Question

If 3 sin x + 5 cos x = 5, then write the value of 5 sin x − 3 cos x.

Answer 1

$$\begin{gathered} 3\sin x+5\cos x=5\left(\mathrm{Given}\right) \\ \mathrm{Squaring}\mathrm{both}\mathrm{the}\mathrm{sides}: \\ 9{\sin }^{2}x+25{\cos }^{2}x+30\sin x\cos x=25 \\ 30\sin x\cos x=25-9{\sin }^{2}x-25{\cos }^{2}x\left(1\right) \\ \mathrm{We}\mathrm{have}\mathrm{to}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}5\sin \mathrm{\theta }-3\cos \mathrm{\theta }. \\ {\left(5\sin \mathit{}x-3\cos \mathrm{x}\right)}^2=25{\sin }^2\mathrm{x}+9{\cos }^{2}x-30\sin x\cos x \\ {\left(5\sin x-3\cos x\right)}^2=25{\sin }^{2}x+9{\cos }^{2}x-\left(25-9{\sin }^{2}x-25{\cos }^{2}x\right)\left[\mathrm{From}\left(1\right)\right] \\ {\left(5\sin x-3\cos x\right)}^2=34{\sin }^{2}x+34{\cos }^{2}x-25 \\ {\left(5\sin x-3\cos x\right)}^2=34-25\left(\because {\sin }^{2}x+{\cos }^{2}x=1\right) \\ {\left(5\sin x-3\cos x\right)}^2=9 \\ 5\sin x-3\cos x=\pm 3 \end{gathered}$$