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a, \(sin\alpha=\frac{1}{5},\frac{\pi}{2}< \alpha< \pi\)
+) \(sin^2\alpha+cos^2\alpha=1\)
\(\Leftrightarrow\left(\frac{1}{5}\right)^2+cos^2\alpha=1\Leftrightarrow cos^2\alpha=\frac{24}{25}\Leftrightarrow cos\alpha=\pm\frac{2\sqrt{6}}{5}\)
mà \(\frac{\pi}{2}< \alpha< \pi\Rightarrow cos\alpha=-\frac{2\sqrt{6}}{5}\)
+) \(tan\alpha=\frac{sin\alpha}{cos\alpha}=\frac{\frac{1}{5}}{-\frac{2\sqrt{6}}{5}}=-\frac{\sqrt{6}}{12}\)
+) \(cot\alpha=\frac{cos\alpha}{sin\alpha}=\frac{-\frac{2\sqrt{6}}{5}}{\frac{1}{5}}=-2\sqrt{6}\)
a/ \(\frac{\pi}{2}< a< \pi\Rightarrow cosa< 0\)
\(\Rightarrow cosa=-\sqrt{1-sin^2a}=-\frac{2\sqrt{6}}{5}\)
\(tanx=\frac{sinx}{cosx}=-\frac{\sqrt{6}}{12}\) ; \(cotx=\frac{1}{tanx}=-2\sqrt{6}\)
b/ \(\frac{3\pi}{2}< a< 2\pi\Rightarrow cosa>0\)
\(\Rightarrow cosa=\frac{1}{\sqrt{1+tan^2a}}=\frac{5\sqrt{26}}{26}\)
\(sina=tana.cosa=-\frac{\sqrt{26}}{26}\)
c/ \(0< a< \frac{\pi}{2}\Rightarrow sina;cosa>0\)
\(\left\{{}\begin{matrix}cos^2a+sin^2a=1\\2sina.cosa=\frac{2}{3}\end{matrix}\right.\)
\(\Rightarrow sina+cosa=\frac{\sqrt{15}}{3}\Rightarrow cosa=\frac{\sqrt{15}}{3}-sina\)
\(\Rightarrow sina\left(\frac{\sqrt{15}}{3}-sina\right)=\frac{1}{3}\Rightarrow sin^2a-\frac{\sqrt{15}}{3}sina+\frac{1}{3}=0\)
\(\Rightarrow\left[{}\begin{matrix}sina=\frac{\sqrt{15}+\sqrt{3}}{6}\Rightarrow cosa=\frac{\sqrt{15}-\sqrt{3}}{6}\\sina=\frac{\sqrt{15}-\sqrt{3}}{6}\Rightarrow cosa=\frac{\sqrt{15}+\sqrt{3}}{6}\end{matrix}\right.\) \(\Rightarrow tana=\frac{sina}{cosa}=...\)
d/ \(\frac{\pi}{2}< a< \pi\Rightarrow\left\{{}\begin{matrix}sina>0\\cosa< 0\end{matrix}\right.\)
\(cosa=\sqrt{2}-sina\) \(\Rightarrow sin^2a+\left(\sqrt{2}-sina\right)^2=1\)
\(\Leftrightarrow2sin^2a-2\sqrt{2}sina+1=0\Rightarrow sina=\frac{\sqrt{2}}{2}\)
\(\Rightarrow cosa=-\sqrt{1-sin^2a}=-\frac{\sqrt{2}}{2}\)
\(tana=\frac{sina}{cosa}=-1\)
\(\left(sina-cosa\right)^2=2\Leftrightarrow sin^2a+cos^2a-2sina.cosa=2\)
\(\Leftrightarrow1-sin2a=2\Rightarrow sin2a=-1\)
\(\left(sina+cosa\right)^2=2\Leftrightarrow sin^2a+cos^2a+2sina.cosa=2\)
\(\Leftrightarrow1+sin2a=2\Rightarrow sin2a=1\)
\(\frac{3\pi}{2}< a< 2\pi\Rightarrow cosa>0\Rightarrow cosa=\sqrt{1-sin^2a}=\frac{1}{2}\)
\(\Rightarrow cos\left(a+\frac{\pi}{3}\right)=cosa.cos\frac{\pi}{3}-sina.sin\frac{\pi}{3}\)
\(=\frac{1}{2}.\frac{1}{2}-\left(-\frac{\sqrt{3}}{2}\right).\left(\frac{\sqrt{3}}{2}\right)=...\)
vậy thì kết quả là
\(\sin2\alpha=-0.96\)
\(\)còn \(\cos\left(\alpha+\frac{\pi}{6}\right)\) thì đúng vì -(-0.8) mà sorry thiếu ngủ hôm qua -_-
Bài 4:
$\sin a=\frac{1}{2}$ và $0< a< \pi$ nên $a=\frac{\pi}{6}$ hoặc $a=\frac{5}{6}\pi$
Nếu $a=\frac{\pi}{6}$ thì $\tan (2a-\frac{\pi}{2})+\sin a=\tan (2.\frac{\pi}{6}-\frac{\pi}{2})+\frac{1}{2}=\frac{-\sqrt{3}}{3}+\frac{1}{2}=\frac{3-2\sqrt{3}}{6}$
Nếu $a=\frac{5\pi}{6}$ thì:
\(\tan (2a-\frac{\pi}{2})+\sin a=\tan (2.\frac{5\pi}{6}-\frac{\pi}{2})+\frac{1}{2}=\frac{\sqrt{3}}{3}+\frac{1}{2}=\frac{3+2\sqrt{3}}{6}\)
Bài 3:
\(\tan a=\frac{-4}{7}=\frac{\sin a}{\cos a}\)
\(\Rightarrow \frac{\sin ^2a}{\cos ^2a}=\frac{16}{49}\Rightarrow \frac{1}{\cos ^2a}=\frac{65}{49}\) \(\Rightarrow \cos ^2a=\frac{49}{65}\)
Kết hợp điều kiện của $a$ suy ra $\cos a>0\Rightarrow \cos a=\frac{7}{\sqrt{65}}$
$\Rightarrow \sin a=\frac{-4}{7}\cos a=\frac{-4}{\sqrt{65}}$
Do đó:
\(\cos (2a-\frac{\pi}{2})=\cos 2a.\cos \frac{\pi}{2}+\sin 2a.\sin \frac{\pi}{2}\)
\(=(\cos ^2a-\sin ^2a).0+2\sin a\cos a.1=2\sin a\cos a=2.\frac{-4}{\sqrt{65}}.\frac{7}{\sqrt{65}}=\frac{56}{65}\)
Công thức hạ bậc
\(sin^2a=\frac{1}{2}-\frac{1}{2}cos2a\)
Julian Edward
\(\frac{\pi}{2}< a< \pi\Rightarrow cosa< 0\)
\(\Rightarrow cosa=-\sqrt{1-sin^2a}=-\frac{4}{5}\)
\(P=1-\left[1-cos\left(\frac{\pi}{2}-2a\right)\right]+sin2a-cos2a-6cota\)
\(=sin2a+sin2a-cos2a-6cota\)
\(=2sin2a-cos2a-6cota\)
\(=4sina.cosa-\left(cos^2a-sin^2a\right)-\frac{6cosa}{sina}\) (thay số và bấm máy)
--.-- \(-\pi>-\frac{3}{2}\pi\) mà
Chắc nhầm đề rồi, phải là \(-\pi>a>-\frac{3}{2}\pi\)mới đúng chứ
\(-\pi>a>-\frac{3}{2}\pi\Leftrightarrow\pi>a>\frac{1}{2}\pi\)
\(\cos a=-\frac{4}{5}\Rightarrow\sin a=\frac{3}{5}\)
\(\sin2a=2\sin a.\cos a=2.\frac{3}{5}.\frac{-4}{5}=-\frac{24}{25}\)
\(\cos2a=2\cos^2a-1=\frac{7}{25}\)
\(\sin\left(\frac{5\pi}{2}-a\right)=\sin\left(\frac{\pi}{2}-a\right)=\cos a=-\frac{4}{5}\)
\(\sin\left(a+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}.\frac{3}{5}-\frac{4}{5}.\frac{\sqrt{2}}{2}=-\frac{\sqrt{2}}{10}\)
\(\cos\left(a+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}.\frac{-4}{5}-\frac{\sqrt{2}}{2}.\frac{3}{5}=-\frac{7\sqrt{2}}{10}\)
\(\Rightarrow\tan\left(a+\frac{\pi}{4}\right)=\frac{1}{7}\)
\(\cos^2\left(\frac{a}{2}\right)=\frac{1+\cos a}{2}=\frac{1}{10}\Leftrightarrow\left|\cos\frac{a}{2}\right|=\frac{\sqrt{10}}{10}\)
Mà \(\frac{\pi}{2}>\frac{a}{2}>\frac{\pi}{4}\)
\(\Rightarrow\cos\frac{a}{2}=\frac{\sqrt{10}}{10}\)
\(0< a< \frac{\pi}{2}\Rightarrow cosa>0\)
\(\Rightarrow cosa=\sqrt{1-sin^2a}=\sqrt{1-\frac{1}{3}}=\frac{\sqrt{6}}{3}\)
\(\Rightarrow cos\left(a+\frac{\pi}{3}\right)=cosa.cos\frac{\pi}{3}-sina.sin\frac{\pi}{3}\)
\(=\frac{\sqrt{6}}{3}.\frac{1}{2}-\frac{1}{\sqrt{3}}.\frac{\sqrt{3}}{2}=\frac{\sqrt{6}-3}{6}\)
cos(α+\(\frac{\Pi}{\sqrt{3}}\))= sinα.sin\(\frac{\Pi}{\sqrt{3}}\)-cosα.cos\(\frac{\Pi}{\sqrt{3}}\)
chứ ạ