Question

giúp với ạ

Answer 1

a) 

\(\tan\alpha=\dfrac{1}{3}\Leftrightarrow\dfrac{\sin\alpha}{\cos\alpha}=\dfrac{1}{3}\Rightarrow3\sin\alpha=\cos\alpha\)

Ta có:

 \(\sin^2\alpha+\cos^2\alpha=1\\ \Leftrightarrow\sin^2\alpha+\left(3\sin\alpha\right)^2=1\\ \Leftrightarrow10\sin^2\alpha=1\\ \Leftrightarrow\sin^2\alpha=\dfrac{1}{10}\)

Vì \(\alpha\) là một góc nhọn => \(\alpha>0.\Rightarrow\sin\alpha=\dfrac{\sqrt{10}}{10}=0,3162\)

\(\cos\alpha=3\sin\alpha=\dfrac{3\sqrt{10}}{10}\approx0,9487\)

b) \(\cot\alpha=\dfrac{3}{4}\Leftrightarrow\dfrac{\cos\alpha}{\sin\alpha}=\dfrac{3}{4}\Rightarrow4\cos\alpha=3\sin\alpha\Leftrightarrow\cos\alpha=\dfrac{3}{4}\sin\alpha\)

Ta có:

\(\sin^2\alpha+\cos^2\alpha=1\\ \Leftrightarrow\sin^2\alpha+\left(\dfrac{3}{4}\sin\alpha\right)^2=1\\ \Leftrightarrow\dfrac{25}{16}\sin^2\alpha=1\\ \Leftrightarrow\sin^2\alpha=\dfrac{16}{25}\)

Vì \(\alpha\) là một góc nhọn => \(\alpha>0\) => \(\sin\alpha=\sqrt{\dfrac{16}{25}}=\dfrac{4}{5}\approx0,8\)

\(\cos\alpha=\dfrac{3}{4}\sin\alpha=\dfrac{3}{4}.0,8=\dfrac{3}{5}=0,6\)