1)Cho góc \(\alpha\) thõa mãn \(sin\alpha+cos\alpha=\frac{\sqrt{2}}{2}\) . Tính P = \(tan^2\alpha+cot^2\alpha\)
2)Cho góc \(\alpha\) thõa mãn \(3cos\alpha+2sin\alpha=2\) và \(sin\alpha< 0\) . Tính sin\(\alpha\)
3)Cho góc \(\alpha\) thõa mãn \(\pi< \alpha< \frac{3\pi}{2}\) và \(sin\alpha-2cos\alpha=1\) . Tính P = \(2tan\alpha-cot\alpha\)
Câu 1:
\(sina+cosa=\frac{\sqrt{2}}{2}\Leftrightarrow\left(sina+cosa\right)^2=\frac{1}{2}\)
Chia 2 vế cho \(cos^2a:\) :
\(\left(\frac{sina+cosa}{cosa}\right)^2=\frac{1}{2}.\frac{1}{cos^2a}\Leftrightarrow\left(tana+1\right)^2=\frac{1}{2}\left(1+tan^2a\right)\)
\(\Leftrightarrow tan^2a+4tana+1=0\)
Tiếp tục chia 2 vế cho \(tana\): :
\(\Rightarrow tana+4+cota=0\Rightarrow tana+cota=-4\)
\(P=tan^2a+cot^2a=tan^2a+2+cot^2a-2=\left(tana+cota\right)^2-2=\left(-4\right)^2-2=14\)
Câu 2:
\(3cosa+2sina=2\Rightarrow cosa=\frac{2-2sina}{3}=\frac{2}{3}\left(1-sina\right)\)
Mặt khác ta luôn có: \(sin^2a+cos^2a=1\Leftrightarrow sin^2a+\frac{4}{9}\left(1-sina\right)^2=1\)
\(\Leftrightarrow9sin^2a+4sin^2a-8sina+4=9\)
\(\Leftrightarrow13sin^2a-8sina-5=0\Rightarrow\left[{}\begin{matrix}sina=1>0\left(l\right)\\sina=-\frac{5}{13}\end{matrix}\right.\)
Câu 3:
\(\left(sina-2cosa\right)^2=1\Leftrightarrow sin^2a-4sina.cosa-4cos^2a=1\)
Chia 2 vế cho \(cos^2a\)
\(tan^2a-4tana+4=\frac{1}{cos^2a}=1+tan^2a\)
\(\Leftrightarrow4tana=3\Rightarrow tana=\frac{3}{4}\)
\(P=2tana-\frac{1}{tana}=2.\frac{3}{4}-\frac{4}{3}=\frac{1}{6}\)