[tex]\sin x+\cos x = \dfrac{3}{5}\Big|^2 \Rightarrow \sin^2 x+2\sin x\cos x+\cos^2 x = \dfrac{9}{25}\\ \\ \Rightarrow 2\sin x\cos x+1 = \dfrac{9}{25} \Rightarrow\sin x \cos x = -\dfrac{8}{25}\\ \\ \\ \sin^3 x+\cos^3 x = \\ \\= (\sin x+\cos x)(\sin^2 x-\sin x\cos x+\cos^2 x)\\ \\=\dfrac{3}{5}\cdot \Big(1+\dfrac{8}{25}\Big)\\\\ =\dfrac{3}{5}\cdot \dfrac{33}{25}\\ \\ = \dfrac{99}{125}[/tex]
