a, \(M=\dfrac{sin\alpha+2cos\alpha}{3sin\alpha-cos\alpha}\)
\(M=\dfrac{\dfrac{sin\alpha}{cos\alpha}+2}{3\dfrac{sin\alpha}{cos\alpha}-1}=\dfrac{tan2+2}{3tan2-1}\)
\(M=\dfrac{3+2}{3.3+1}=\dfrac{5}{8}\)
câu b mẫu =0 , lỗi á
c, \(tan\alpha=\dfrac{sin\alpha}{cos\alpha}=3\Rightarrow sin\alpha=3cos\alpha\)
\(P=\dfrac{sin^2\alpha+2sin\alpha cos\alpha+3}{2sin^2\alpha-sin\alpha cos\alpha+cos^2\alpha+1}\)
\(P=\dfrac{9cos^2\alpha+2.3cos\alpha cos\alpha+3}{2.9cos^2\alpha-3cos\alpha cos\alpha+1}\)
\(P=\dfrac{15cos^2\alpha+3}{15cos^2\alpha+1}=\dfrac{15cos^2\alpha+1}{15cos^2\alpha+1}+\dfrac{2}{15cos^2\alpha+1}\)
\(P=1+\dfrac{2}{15cos^2\alpha+1}\)
ta có: \(1+tan^2\alpha=\dfrac{1}{cos^2\alpha}\Rightarrow1+3^2=\dfrac{1}{cos^2\alpha+1}\)
\(\Rightarrow cos^2\alpha=\dfrac{1}{10}\)
\(\Rightarrow P=1+\dfrac{2}{15.\dfrac{1}{10}+1}=\dfrac{9}{5}\)
d, \(Q=\dfrac{2sin^2\alpha+sin\alpha cos\alpha+3}{2sin^2\alpha-2sin\alpha cos\alpha+2cos^2\alpha}\)
ta có : \(tan\alpha=\dfrac{sin\alpha}{cos\alpha}=3\Rightarrow sin\alpha=3cos\alpha\)
\(Q=\dfrac{2.9cos^2\alpha+3cos\alpha cos\alpha+1}{9cos^2\alpha-2.3cos\alpha cos\alpha+2cos^2\alpha}\)
\(Q=\dfrac{18cos^2\alpha+3cos^2\alpha+1}{9cos^2\alpha-6cos^2\alpha+2cos^2\alpha}\)
\(Q=\dfrac{21cos^2\alpha+1}{5cos^2\alpha}=\dfrac{21}{5}+\dfrac{1}{5cos^2\alpha}\)
ta có : \(1+tan^2\alpha=\dfrac{1}{cos^2\alpha}\Rightarrow1+3^2=\dfrac{1}{cos^2\alpha}\)
\(\Rightarrow cos^2\alpha=\dfrac{1}{10}\)
\(Q=\dfrac{21}{5}+\dfrac{1}{5.\dfrac{1}{10}}=\dfrac{31}{5}\)
