Bài 47:
\(tan\alpha+cot\alpha=3\)
=>\(\dfrac{sin\alpha}{cos\alpha}+\dfrac{cos\alpha}{sin\alpha}=3\)
=>\(\dfrac{sin^2\alpha+cos^2\alpha}{sin\alpha\cdot cos\alpha}=3\)
=>\(\dfrac{1}{sin\alpha\cdot cos\alpha}=3\)
=>\(sin\alpha\cdot cos\alpha=\dfrac{1}{3}\)
Bài 48:
a: \(sin^2\beta+cos^2\beta=1\)
=>\(cos^2\beta=1-\left(\dfrac{1}{4}\right)^2=\dfrac{15}{16}\)
=>\(cos\beta=\dfrac{\sqrt{15}}{4}\)
\(tan\beta=\dfrac{sin\beta}{cos\beta}=\dfrac{1}{4}:\dfrac{\sqrt{15}}{4}=\dfrac{1}{\sqrt{15}}\)
b:
\(cos^2\alpha+sin^2\alpha=1\)
=>\(sin^2\alpha=1-\left(\dfrac{1}{3}\right)^2=\dfrac{8}{9}\)
=>\(sin\alpha=\sqrt{\dfrac{8}{9}}=\dfrac{2\sqrt{2}}{3}\)
\(tan\alpha=\dfrac{sin\alpha}{cos\alpha}=\dfrac{2\sqrt{2}}{3}:\dfrac{1}{3}=2\sqrt{2}\)
\(cot\alpha=\dfrac{1}{tan\alpha}=\dfrac{1}{2\sqrt{2}}=\dfrac{\sqrt{2}}{4}\)
c: \(1+tan^2\beta=\dfrac{1}{cos^2\beta}\)
=>\(\dfrac{1}{cos^2\beta}=1+\left(\sqrt{3}\right)^2=4\)
=>\(cos^2\beta=\dfrac{1}{4}\)
=>\(cos\beta=\sqrt{\dfrac{1}{4}}=\dfrac{1}{2}\)
\(tan\beta=\dfrac{sin\beta}{cos\beta}\)
=>\(sin\beta=\dfrac{1}{2}\cdot\sqrt{3}=\dfrac{\sqrt{3}}{2}\)
