Solution-1

Divide both sides of the equation by \(5\):

\[\frac{3}{5} \sin x+\frac{4}{5} \cos x=\frac{6}{5}.\]

Choose an acute angle \(\alpha\) such that \(\sin \alpha=\frac{4}{5}\). In this case \(\cos \alpha = \sqrt{1-\sin ^{2} \alpha} = \sqrt{1-\left(\frac{4}{5}\right)^{2}}=\frac{3}{5}\) and the equation becomes

\[\begin{gathered} \sin x \cos \alpha+\cos x \sin \alpha=\frac{6}{5} \\ \sin (x+\alpha)=\frac{6}{5} \end{gathered}\]

Sine values never exceed \(1\), so there are no solutions for this equation. And there is no solution for the original equation either.

Atrisinājums-1

Izdalām abas vienādojuma puses ar \(5\):

\[\frac{3}{5} \sin x+\frac{4}{5} \cos x=\frac{6}{5}.\]

Izvēlēsimies tādu šauru leņki \(\alpha\), ka \(\sin \alpha=\frac{4}{5}\). Tādā gadījumā \(\cos \alpha=\sqrt{1-\sin ^{2} \alpha}=\sqrt{1-\left(\frac{4}{5}\right)^{2}}=\frac{3}{5}\) un vienādojumu varam pārrakstīt kā

\[\begin{aligned} & \sin x \cos \alpha+\cos x \sin \alpha=\frac{6}{5} \\ & \sin (x+\alpha)=\frac{6}{5} \end{aligned}\]

Tā kā sinusa vērtības nepārsniedz 1, tad šim vienādojumam atrisinājumu nav. Tātad arī dotajam vienādojumam atrisinājuma nav.