Question

Cho gócα thỏa mãn cot\(\)α=\(\dfrac{2}{3}\),Tính giá trị biểu thức P=\(\dfrac{sin^2a+1}{2.sina.cosa}\)bằng bao nhiêu

Answer 1

\(1+\cot^2a=\dfrac{1}{\sin^2a}=1+\dfrac{4}{9}=\dfrac{13}{9}\)

\(\Leftrightarrow\sin^2a=\dfrac{9}{13}\)

\(\Leftrightarrow\cos^2a=\dfrac{4}{13}\)

\(P=\dfrac{\sin^2a+1}{2\cdot\sin a\cdot\cos a}=\dfrac{22}{13}:\left(2\cdot\sin a\cdot\cos a\right)\)

Trường hợp 1: \(\sin a;\cos a\) cùng dấu

=>\(\sin a\cdot\cos a=\dfrac{3\sqrt{13}}{13}\cdot\dfrac{2\sqrt{13}}{13}=\dfrac{6\cdot13}{169}=\dfrac{6}{13}\)

\(\Leftrightarrow P=\dfrac{22}{13}:\dfrac{12}{13}=\dfrac{22}{12}=\dfrac{11}{6}\)

Trường hợp 2: \(\sin a;\cos a\) khác dấu

\(\Leftrightarrow P=\dfrac{22}{13}:\dfrac{-12}{13}=\dfrac{-11}{6}\)