Bài 4 :
a) \(cos\alpha=0,8\Rightarrow cos^2\alpha=0,64\)
\(sin^2\alpha+cos^2\alpha=1\Rightarrow sin^2\alpha=1-cos^2\alpha\)
\(\Rightarrow sin^2\alpha=1-0,64=0,36\)
\(\Rightarrow sin\alpha=0,6\)
\(tan\alpha=\dfrac{sin\alpha}{cos\alpha}=\dfrac{0,6}{0,8}=0,75\)
\(cot\alpha=\dfrac{1}{tan\alpha}=\dfrac{1}{0,75}=\dfrac{1}{\dfrac{3}{4}}=\dfrac{4}{3}\)
b) \(tan\alpha=\dfrac{3}{4}\)
\(1+tan^2\alpha=\dfrac{1}{cos^2\alpha}\)
\(\Rightarrow cos^2\alpha=\dfrac{1}{1+tan^2\alpha}=\dfrac{1}{1+\dfrac{9}{16}}=\dfrac{16}{25}\)
\(\Rightarrow cos\alpha=\dfrac{4}{5}\)
\(sin^2\alpha+cos^2\alpha=1\Rightarrow sin^2\alpha=1-cos^2\alpha\)
\(\Rightarrow sin^2\alpha=1-\dfrac{16}{25}=\dfrac{9}{25}\)
\(\Rightarrow sin\alpha=\dfrac{3}{5}\)
\(tan\alpha=\dfrac{sin\alpha}{cos\alpha}=\dfrac{\dfrac{3}{5}}{\dfrac{4}{5}}=\dfrac{3}{4}\)
\(cot\alpha=\dfrac{1}{tan\alpha}=\dfrac{1}{\dfrac{3}{4}}=\dfrac{4}{3}\)
Bài 6 : \(tanB=2\)
a) \(A=\dfrac{sinB+cosB}{sinB-cosB}\)
\(\Leftrightarrow A=\dfrac{\dfrac{sinB}{cosB}+\dfrac{cosB}{cosB}}{\dfrac{sinB}{cosB}-\dfrac{cosB}{cosB}}\)
\(\Leftrightarrow A=\dfrac{tanB+1}{tanB-1}=\dfrac{2+1}{2-1}=3\)
b) \(B=\dfrac{2sinB+cosB}{3sinB-4cosB}\)
\(\Leftrightarrow B=\dfrac{2\dfrac{sinB}{cosB}+\dfrac{cosB}{cosB}}{\dfrac{3sinB}{cosB}-\dfrac{4cosB}{cosB}}\)
\(\Leftrightarrow B=\dfrac{2tanB+1}{3tanB-4}=\dfrac{2.2+1}{3.2-4}=\dfrac{5}{2}\)
c) \(C=sin^2B-2sinB.cosB-3cos^2B\)
\(\Leftrightarrow C=tan^2B.cos^2B-2tanB.cos^2B-3cos^2B\)
\(\Leftrightarrow C=cos^2B.\left(tan^2B-2tanB-3\right)\)
\(\Leftrightarrow C=\dfrac{1}{1+tan^2B}.\left(tan^2B-2tanB-3\right)\)
\(\Leftrightarrow C=\dfrac{1}{1+4}.\left(4-2.2-3\right)=-\dfrac{3}{5}\)
d) \(D=\dfrac{sin^2B-sinB.cosB-cos^2B}{2sinB.cóB}\)
\(\Leftrightarrow D=tanB-1-cotB\)
\(\Leftrightarrow D=tanB-1-\dfrac{1}{tanB}\)
\(\Leftrightarrow D=2-1-\dfrac{1}{2}=\dfrac{1}{2}\)
