+26367 If
\(\sin(x) + \cos(x) = \dfrac{1}{2}\),
find
\(\sin^3(x) + \cos^3(x)\).
\(\begin{array}{|rcll|} \hline \Big( \sin(x) + \cos(x) \Big)^2 &=& \sin^2(x) + 2\sin(x)\cos(x)+\cos^2(x) \\ \Big( \sin(x) + \cos(x) \Big)^2 &=& \sin^2(x)+\cos^2(x) + 2\sin(x)\cos(x) \quad | \quad \sin^2(x)+\cos^2(x)=1 \\ \Big( \sin(x) + \cos(x) \Big)^2 &=& 1 + 2\sin(x)\cos(x) \quad | \quad \sin(x) + \cos(x) = \dfrac{1}{2} \\ \Big(\dfrac{1}{2} \Big)^2 &=& 1 + 2\sin(x)\cos(x) \\ 2\sin(x)\cos(x) &=& \dfrac{1}{4} - 1 \\ 2\sin(x)\cos(x) &=& -\dfrac{3}{4} \\ \mathbf{ \sin(x)\cos(x) } &=& \mathbf{-\dfrac{3}{8} } \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \Big(\sin^2(x) + \cos^2(x)\Big)\Big( \sin(x) + \cos(x) \Big) &=& \sin^3(x) +\sin^2(x)\cos(x) +\cos^2(x)\sin(x) + \cos^3(x) \\ \Big(\sin^2(x) + \cos^2(x)\Big)\Big( \sin(x) + \cos(x) \Big) &=& \sin^3(x)+\cos^3(x) +\sin(x)\cos(x)\Big(\sin(x) +\cos(x)\Big) \\ 1*\dfrac{1}{2} &=& \sin^3(x)+\cos^3(x) -\dfrac{3}{8}*\dfrac{1}{2} \\ \dfrac{1}{2} &=& \sin^3(x)+\cos^3(x) -\dfrac{3}{16} \\ \sin^3(x)+\cos^3(x) &=& \dfrac{3}{16}+\dfrac{1}{2} \\ \mathbf{ \sin^3(x)+\cos^3(x) } &=& \mathbf{ \dfrac{11}{16} } \\ \hline \end{array}\)
