<s>[INST] If $\sin \alpha +\cos \alpha =1.2$, then what is $\sin^3\alpha + \cos^3\alpha$? If $\sin \alpha +\cos \alpha =1.2$, then what is $\sin^3\alpha + \cos^3\alpha$?

All I know is that $\sin^{3}a+\cos^{3}a$ is equal to 
$$(\sin a + \cos a)(\sin^{2} a - \sin a \cos a + \cos^{2} a)= \dfrac{6}{5}(\sin^{2} a - \sin a \cos a + \cos^{2} a)$$  But now, I'm stuck.  Solutions are greatly appreciated.
 [/INST] $$\sin^3 \alpha+\cos^3 \alpha=(\sin \alpha+\cos \alpha)(\sin^2 \alpha-\sin \alpha\cos \alpha+\cos^2 \alpha)$$
$$1.2^3=\sin^3\alpha+\cos^3 \alpha+3\sin \alpha\cos \alpha(\sin \alpha+\cos \alpha)$$ It follows $$\sin \alpha\cos \alpha=0.22$$ and because $$\sin \alpha+\cos \alpha=1.2$$ one has sum and product of two unknowns so its are solution of the quadratic equation $$X^2-1.2X+0.22=0$$ with positive discriminant $0.14$.
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