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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
\(\sin\alpha=\sqrt{1-\left(\dfrac{20}{29}\right)^2}=\dfrac{21}{29}\)
\(\tan\alpha=\dfrac{21}{20}\)
\(\cot\alpha=\dfrac{20}{21}\)
\(\sin\alpha=\sqrt{1-\dfrac{400}{29^2}}=\dfrac{21}{29}\)
\(\tan\alpha=\dfrac{21}{20}\)
\(\cot\alpha=\dfrac{20}{21}\)
*Ta có \(\cos^2a+\sin^2a=1\)
\(\Rightarrow sina=\sqrt{1-\cos^2a}=\sqrt{1-\left(\dfrac{20}{29}\right)^2}=\dfrac{21}{29}\)
*Ta có \(\tan a=\dfrac{\sin a}{\cos a}=\dfrac{21}{29}:\dfrac{20}{29}=\dfrac{21}{20}\)
*Ta có \(\cot a.\tan a=1\Rightarrow\cot a=\dfrac{1}{\tan a}=\dfrac{20}{21}\)
1. Ta có \(1+\tan\alpha=\dfrac{1}{\cos^2\alpha}\Rightarrow\dfrac{1}{\cos^2\alpha}=1+\dfrac{1}{3}\Rightarrow\dfrac{1}{\cos^2\alpha}=\dfrac{4}{3}\Rightarrow\cos^2\alpha=\dfrac{3}{4}\Rightarrow\cos\alpha=\dfrac{\sqrt{3}}{2}\)
Mặt khác, \(tan\alpha=\dfrac{1}{3}=\dfrac{\sin\alpha}{\cos\alpha}\Rightarrow\sin\alpha=\dfrac{\cos a}{3}=\dfrac{\dfrac{\sqrt{3}}{2}}{3}=\dfrac{1}{2\sqrt{3}}\)
2. Ta có \(1+\cot^2\alpha=\dfrac{1}{\sin^2\alpha}\Rightarrow\dfrac{1}{\sin^2\alpha}=1+\dfrac{9}{16}\Rightarrow\dfrac{1}{\sin^2\alpha}=\dfrac{25}{16}\Rightarrow\dfrac{1}{\sin a}=\dfrac{5}{4}\Rightarrow\sin\alpha=\dfrac{4}{5}\)
Mặt khác, \(\cot\alpha=\dfrac{\cos\alpha}{\sin\alpha}\Rightarrow\cos\alpha=\sin\alpha.\cot\alpha=\dfrac{3}{4}.\dfrac{4}{5}=\dfrac{3}{5}\)
\(\sin a=\frac{\sqrt{3}}{2}\Rightarrow\widehat{a}=60^0\)
\(\Rightarrow\cos a=\cos60^0=\frac{1}{2}\)
\(\tan a=\tan60^0=\sqrt{3}\)
\(\cot a=\frac{1}{\tan a}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\)
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
\(sina=\sqrt{1-cos^2a}=\dfrac{2}{\sqrt{5}}\)