Question

Cho \(0< \alpha< 90\) độ. Không dùng máy tính hãy tính :

\(\sin\alpha+\cos\alpha,\) biết \(\tan\alpha+\cot\alpha=8\)

Answer 1

Có \(\tan\alpha=\frac{\sin\alpha}{\cos\alpha};\cot\alpha=\frac{\cos\alpha}{\sin\alpha}\)

\(\Rightarrow\frac{\sin\alpha}{\cos\alpha}+\frac{\cos\alpha}{\sin\alpha}=8\)

\(\Leftrightarrow\sin^2\alpha+\cos^2\alpha=8\sin\alpha.\cos\alpha\)

\(\Leftrightarrow\sin\alpha.\cos\alpha=\frac{1}{8}\Leftrightarrow2\sin\alpha.\cos\alpha=\frac{1}{4}\)

Có \(\sin^2\alpha+\cos^2\alpha=1\Leftrightarrow\sin^2\alpha+2\sin\alpha.\cos\alpha+\cos^2\alpha=1+2\sin\alpha.\cos\alpha\)

\(\Leftrightarrow\left(\sin\alpha+\cos\alpha\right)^2=1+\frac{1}{4}=\frac{5}{4}\)

\(\Leftrightarrow\sin\alpha+\cos\alpha=\frac{\sqrt{5}}{2}\)