Question

Cho sin α + cos α= 2 phần 5 tính P= sin^3 α+ cos^3 α

Answer 1

Có \(sin\alpha+cos\alpha=\dfrac{2}{5}\Leftrightarrow\left(sin\alpha+cos\alpha\right)^2=\dfrac{4}{25}\)

\(\Leftrightarrow sin^2\alpha+2sin\alpha\cdot cos\alpha+cos^2\alpha=\dfrac{4}{25}\)

\(\Leftrightarrow\left(sin^2\alpha+cos^2\alpha\right)+2sin\alpha\cdot cos\alpha=\dfrac{4}{25}\)

\(\Leftrightarrow1+2sin\alpha\cdot cos\alpha=\dfrac{4}{25}\Leftrightarrow sin\alpha\cdot cos\alpha=-\dfrac{21}{50}\)

Ta có:

\(P=sin^3\alpha+cos^3\alpha=\left(sin\alpha+cos\alpha\right)\left(sin^2\alpha-sin\alpha\cdot cos\alpha+cos^2\alpha\right)\)

     \(=2\left[1-\left(-\dfrac{21}{50}\right)\right]=\dfrac{71}{25}\)