cho sinx +cosx=5/4 tính sinx , cosx

Đáp án:

\[\left[ \begin{array}{l}
\cos x = \frac{{5 - \sqrt 7 }}{8};\sin x = \frac{{5 + \sqrt 7 }}{8}\\
\cos x = \frac{{5 + \sqrt 7 }}{8};\sin x = \frac{{5 - \sqrt 7 }}{8}
\end{array} \right.\]

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
\sin x + \cos x = \frac{5}{4} \Leftrightarrow \sin x = \frac{5}{4} - \cos x\\
{\sin ^2}x + {\cos ^2}x = 1\\
 \Leftrightarrow {\left( {\frac{5}{4} - \cos x} \right)^2} + {\cos ^2}x = 1\\
 \Leftrightarrow \frac{{25}}{{16}} - \frac{5}{2}\cos x + {\cos ^2}x + {\cos ^2}x = 1\\
 \Leftrightarrow 2{\cos ^2}x - \frac{5}{2}\cos x + \frac{9}{{16}} = 0\\
 \Leftrightarrow \left[ \begin{array}{l}
\cos x = \frac{{5 - \sqrt 7 }}{8} \Rightarrow \sin x = \frac{{5 + \sqrt 7 }}{8}\\
\cos x = \frac{{5 + \sqrt 7 }}{8} \Rightarrow \sin x = \frac{{5 - \sqrt 7 }}{8}
\end{array} \right.
\end{array}\)