Question

Cho Sin³a+ Cos³a= \(\frac{5\sqrt{2}}{8}\) . Tính P= Sin a + Cos a

Answer 1

\(sin^3a+cos^3a=\left(sina+cosa\right)\left(1-sina.cosa\right)=\frac{5\sqrt{2}}{8}\)

\(P=sina+cosa\Rightarrow P^2=1+2sina.cosa\Rightarrow sina.cosa=\frac{P^2-1}{2}\)

\(\Rightarrow P\left(1-\frac{P^2-1}{2}\right)=\frac{5\sqrt{2}}{8}\)

\(\Leftrightarrow4P^3-12P+5\sqrt{2}=0\)

\(\Leftrightarrow\left(2P-\sqrt{2}\right)\left(2P^2+\sqrt{2}P-5\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}P=\frac{\sqrt{2}}{2}\\P=\frac{\sqrt{42}-\sqrt{2}}{4}\\P=\frac{-\sqrt{42}-\sqrt{2}}{4}< -\sqrt{2}\left(l\right)\end{matrix}\right.\)