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4.
Pt có 2 nghiệm pb dương khi và chỉ khi:
\(\left\{{}\begin{matrix}\Delta=m^2-4\left(5-m\right)>0\\x_1+x_2=m>0\\x_1x_2=5-m>0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m^2+4m-20>0\\m>0\\m< 5\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}m< -2-2\sqrt{6}\\m>-2+2\sqrt{6}\end{matrix}\right.\\m>0\\m< 5\end{matrix}\right.\) \(\Leftrightarrow-2+2\sqrt{6}< m< 5\)
a.
\(90^0< a< 180^0\Rightarrow cosa< 0\)
\(\Rightarrow cosa=-\sqrt{1-sin^2a}=-\dfrac{2\sqrt{2}}{3}\)
\(tana=\dfrac{sina}{cosa}=-\dfrac{\sqrt{2}}{4}\)
b.
\(0< a< 90^0\Rightarrow cosa>0\)
\(\Rightarrow cosa=\sqrt{1-sin^2a}=\dfrac{4}{5}\)
\(tana=\dfrac{sina}{cosa}=\dfrac{3}{4}\)
\(cota=\dfrac{1}{tana}=\dfrac{4}{3}\)
c.
\(A=\dfrac{\dfrac{sina}{cosa}+\dfrac{3cosa}{sina}}{\dfrac{sina}{cosa}+\dfrac{cosa}{sina}}=\dfrac{sin^2a+3cos^2a}{sin^2a+cos^2a}=1+2cos^2a=\dfrac{17}{8}\)
d.
\(A=\dfrac{\dfrac{cosa}{sina}+\dfrac{3sina}{cosa}}{\dfrac{2cosa}{sina}+\dfrac{sina}{cosa}}=\dfrac{cos^2a+3sin^2a}{2cos^2a+sin^2a}=\dfrac{cos^2a+3\left(1-cos^2a\right)}{2cos^2a+\left(1-cos^2a\right)}\)
\(=\dfrac{3-2cos^2a}{1+cos^2a}=\dfrac{19}{13}\)
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