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\(tana=\sqrt{3}\)
=>\(\dfrac{sina}{cosa}=\sqrt{3}\)
=>\(sina=\sqrt{3}\cdot cosa\)
\(1+tan^2a=\dfrac{1}{cos^2a}\)
=>\(\dfrac{1}{cos^2a}=1+3=4\)
=>\(cos^2a=\dfrac{1}{4}\)
=>\(cosa=\dfrac{1}{2}\)
=>\(sina=\dfrac{\sqrt{3}}{2}\)
\(A=\dfrac{sin^2a-cos^2a}{sina\cdot cosa}\)
\(=\dfrac{\dfrac{3}{4}-\dfrac{1}{4}}{\dfrac{\sqrt{3}}{2}\cdot\dfrac{1}{2}}=\dfrac{2}{4}:\dfrac{\sqrt{3}}{4}=\dfrac{2}{\sqrt{3}}=\dfrac{2\sqrt{3}}{3}\)
1:
a: sin a=căn 3/2
\(cosa=\sqrt{1-sin^2a}=\sqrt{1-\dfrac{3}{4}}=\sqrt{\dfrac{1}{4}}=\dfrac{1}{2}\)
\(tana=\dfrac{\sqrt{3}}{2}:\dfrac{1}{2}=\sqrt{3}\)
cot a=1/tan a=1/căn 3
b: \(tana=2\)
=>cot a=1/tan a=1/2
\(1+tan^2a=\dfrac{1}{cos^2a}\)
=>\(\dfrac{1}{cos^2a}=5\)
=>cos^2a=1/5
=>cosa=1/căn 5
\(sina=\sqrt{1-cos^2a}=\sqrt{\dfrac{4}{5}}=\dfrac{2}{\sqrt{5}}\)
c: \(cosa=\sqrt{1-\left(\dfrac{5}{13}\right)^2}=\dfrac{12}{13}\)
tan a=5/13:12/13=5/12
cot a=1:5/12=12/5
a: sin a=2/3
=>cos^2a=1-(2/3)^2=5/9
=>\(cosa=\dfrac{\sqrt{5}}{3}\)
\(tana=\dfrac{2}{3}:\dfrac{\sqrt{5}}{3}=\dfrac{2}{\sqrt{5}}\)
\(cota=1:\dfrac{2}{\sqrt{5}}=\dfrac{\sqrt{5}}{2}\)
b: cos a=1/5
=>sin^2a=1-(1/5)^2=24/25
=>\(sina=\dfrac{2\sqrt{6}}{5}\)
\(tana=\dfrac{2\sqrt{6}}{5}:\dfrac{1}{5}=2\sqrt{6}\)
\(cota=\dfrac{1}{2\sqrt{6}}=\dfrac{\sqrt{6}}{12}\)
c: cot a=1/tana=1/2
\(1+tan^2a=\dfrac{1}{cos^2a}\)
=>1/cos^2a=1+4=5
=>cos^2a=1/5
=>cosa=1/căn 5
\(sina=\sqrt{1-cos^2a}=\dfrac{2}{\sqrt{5}}\)
\(A=\dfrac{\left(sina+cosa\right)\left(sin^2a-sina\cdot cosa+cos^2a\right)}{cosa\cdot sina\left(2cosa+sina\right)}\)
\(=\dfrac{\left(sina+cosa\right)\left(1-sina\cdot cosa\right)}{cosa\cdot sina\left(2\cdot cosa+sina\right)}\)
\(1+tan^2a=\dfrac{1}{cos^2a}=1+\dfrac{9}{25}=\dfrac{34}{25}\)
\(\Leftrightarrow cosa=\dfrac{5}{\sqrt{34}}\)
=>\(sina=\dfrac{3}{\sqrt{34}}\)
\(=\dfrac{\left(sina+cosa\right)\left(1-sina\cdot cosa\right)}{cosa\cdot sina\left(2\cdot cosa+sina\right)}\)
\(=\dfrac{\left[\left(\dfrac{3}{\sqrt{34}}+\dfrac{5}{\sqrt{34}}\right)\left(1-\dfrac{15}{34}\right)\right]}{\dfrac{15}{34}\cdot\left(\dfrac{10}{\sqrt{34}}+\dfrac{3}{\sqrt{34}}\right)}\)
\(=\dfrac{\dfrac{8}{\sqrt{34}}\cdot\dfrac{19}{34}}{\dfrac{15}{34}\cdot\dfrac{13}{\sqrt{34}}}=\dfrac{8\cdot19}{15\cdot13}=\dfrac{152}{195}\)