\(a\sin x+b\cos x=c\)
\(a^2\sin^2x+b^2\cos^2x+2ab\sin x\cos x=c^2\)
\((a^2-c^2)\sin^2x+(b^2-c^2)\cos^2x+2ab\sin x\cos x=0\)
\((a^2-c^2)\tan^2x+(b^2-c^2)+2ab\tan x=0\)
\((a^2-c^2)t^2+2abt+(b^2-c^2)=0\)
\(\tan x=\dfrac{ab\pm c\sqrt{a^2+b^2-c^2}}{c^2-a^2}\)

\(a\sin x+b\cos x=c\)
\(a\sin x=c-b\cos x\)
\(a^2\sin^2x=c^2+b^2\cos^2x-2bc\cos x\)
\(a^2-a^2\cos^2x=c^2+b^2\cos^2x-2bc\cos x\)
\((a^2+b^2)\cos^2x-2bc\cos x+(c^2-a^2)=0\)
\((a^2+b^2)t^2-2bct+(c^2-a^2)=0\)
\(\cos x=\dfrac{bc\pm a\sqrt{a^2+b^2-c^2}}{a^2+b^2}\)

\(a\sin x+b\cos x=c\)
\(b\cos x=c-a\sin x\)
\(b^2cos^2x=c^2+a^2\sin^2x-2ac\sin x\)
\(b^2-b^2\sin^2x=c^2+a^2\sin^2x-2ac\sin x\)
\((a^2+b^2)\sin^2x-2ac\sin x+(c^2-b^2)=0\)
\((a^2+b^2)t^2-2act+(c^2-b^2)=0\)
\(\sin x=\dfrac{ac\pm b\sqrt{a^2+b^2-c^2}}{a^2+b^2}\)