Bài tập nâng cao · Bài 14
Bài tập nâng cao Giá trị lượng giác của góc lượng giác · Bài 14
Tính giá trị lượng giác của góc $\alpha$ nếu
a) $\sin\alpha=-\frac{2}{5}\,;\,\,\pi < \alpha < \frac{3\pi}{2}$.
b) $\cos\alpha=0,8\,;\,\,\frac{3\pi}{2} < \alpha < 2\pi$.
c) $\tan\alpha=\frac{13}{8}\,;\,\,0 < \alpha < \frac{\pi}{2}$.
d) $\cot\alpha=-\frac{19}{7}\,;\,\,\frac{\pi}{2} < \alpha < \pi$.
a) $\sin\alpha=-\frac{2}{5}\,;\,\,\pi < \alpha < \frac{3\pi}{2}$.
b) $\cos\alpha=0,8\,;\,\,\frac{3\pi}{2} < \alpha < 2\pi$.
c) $\tan\alpha=\frac{13}{8}\,;\,\,0 < \alpha < \frac{\pi}{2}$.
d) $\cot\alpha=-\frac{19}{7}\,;\,\,\frac{\pi}{2} < \alpha < \pi$.
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Lời giải
a) $\pi < \alpha < \frac{3\pi}{2}$ $\Rightarrow \cos\alpha < 0$ nên $\cos\alpha=-\sqrt{1-\sin^{2}\alpha}=-\frac{\sqrt{21}}{5}$; $\tan\alpha=\frac{2}{\sqrt{21}}$; $\cot\alpha=\frac{\sqrt{21}}{2}$.
b) $\frac{3\pi}{2} < \alpha < 2\pi$ $\Rightarrow \sin\alpha < 0$ nên $\sin\alpha=-\sqrt{1-\cos^{2}\alpha}=-0,6$; $\tan\alpha=-\frac{3}{4}$; $\cot\alpha=-\frac{4}{3}$.
c) $0 < \alpha < \frac{\pi}{2}$ $\Rightarrow \cos\alpha>0$ nên $\cos\alpha=\frac{1}{\sqrt{1+\tan^{2}\alpha}}=\frac{8}{\sqrt{233}}$; $\sin\alpha=\tan\alpha.\cos\alpha=\frac{13}{\sqrt{233}}$; $\cot\alpha=\frac{1}{\tan\alpha}=\frac{8}{13}$.
d) $\frac{\pi}{2} < \alpha < \pi$ $\Rightarrow \sin\alpha>0$ nên $\sin\alpha=\frac{1}{\sqrt{1+\cot^{2}\alpha}}=\frac{7}{\sqrt{410}}$; $\cos\alpha=\cot\alpha.\sin\alpha=-\frac{19}{\sqrt{410}}$; $\tan\alpha=\frac{1}{\cot\alpha}=-\frac{7}{19}$.
b) $\frac{3\pi}{2} < \alpha < 2\pi$ $\Rightarrow \sin\alpha < 0$ nên $\sin\alpha=-\sqrt{1-\cos^{2}\alpha}=-0,6$; $\tan\alpha=-\frac{3}{4}$; $\cot\alpha=-\frac{4}{3}$.
c) $0 < \alpha < \frac{\pi}{2}$ $\Rightarrow \cos\alpha>0$ nên $\cos\alpha=\frac{1}{\sqrt{1+\tan^{2}\alpha}}=\frac{8}{\sqrt{233}}$; $\sin\alpha=\tan\alpha.\cos\alpha=\frac{13}{\sqrt{233}}$; $\cot\alpha=\frac{1}{\tan\alpha}=\frac{8}{13}$.
d) $\frac{\pi}{2} < \alpha < \pi$ $\Rightarrow \sin\alpha>0$ nên $\sin\alpha=\frac{1}{\sqrt{1+\cot^{2}\alpha}}=\frac{7}{\sqrt{410}}$; $\cos\alpha=\cot\alpha.\sin\alpha=-\frac{19}{\sqrt{410}}$; $\tan\alpha=\frac{1}{\cot\alpha}=-\frac{7}{19}$.