Tính các giá trị lượng giác của cung α biết:
a) \(\sin \alpha = 0,6\) khi \(0 < \alpha < \frac{\pi }{2}\)
b) \(\cos \alpha = - 0,7\) khi \(\frac{\pi }{2} < \alpha < \pi \)
c) \(\tan \alpha = 2\) khi \(\pi < \alpha < \frac{{3\pi }}{2}\)
d) \(\cot \alpha = - 3\) khi \(\frac{{3\pi }}{2} < \alpha < 2\pi \)
a) \(0 < \alpha < \frac{\pi }{2}\) \( \Rightarrow \cos \alpha > 0\), do đó:
\(\begin{array}{l}
\cos \alpha = \sqrt {1 - {{\sin }^2}\alpha } = \sqrt {1 - 0,36} = 0,8\\
\Rightarrow \tan \alpha = \frac{3}{4},\cot \alpha = \frac{4}{3}
\end{array}\)
b) \(\frac{\pi }{2} < \alpha < \pi \) \( \Rightarrow \sin\alpha > 0\), do đó:
\(\sin \alpha = \sqrt {1 - {{\cos }^2}\alpha } = \sqrt {1 - 0,49} \approx 0,71\)
\( \Rightarrow \tan \alpha = - \frac{{0,7}}{{0,71}} \approx - 0,98,\cot \alpha \approx - 1,01\)
c) \(\pi < \alpha < \frac{{3\pi }}{2}\) \( \Rightarrow \cos \alpha < 0\), do đó:
\(\cos \alpha = - \frac{1}{{\sqrt {1 + {{\tan }^2}\alpha } }} = - \frac{1}{{\sqrt 5 }} = - \frac{{\sqrt 5 }}{5},\sin \alpha = - \frac{{2\sqrt 5 }}{5},\cot \alpha = \frac{1}{2}\)
d) \(\frac{{3\pi }}{2} < \alpha < 2\pi \) \( \Rightarrow \sin\alpha < 0\), do đó:
\(\sin \alpha = - \frac{1}{{\sqrt {1 + {{\cot }^2}\alpha } }} = - \frac{1}{{\sqrt {10} }} = - \frac{{\sqrt {10} }}{{10}},\sin \alpha = - \frac{{3\sqrt {10} }}{{10}},\tan \alpha = - \frac{1}{3}\)
-- Mod Toán 10