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a/ \(cosa=-\sqrt{1-sin^2a}=-\frac{\sqrt{5}}{3}\)
\(tana=\frac{sina}{cosa}=-\frac{2\sqrt{5}}{5}\) ; \(cota=\frac{1}{tana}=-\frac{\sqrt{5}}{2}\)
b/ \(\frac{1}{cos^2a}=1+tan^2a\Rightarrow cos^2a=\frac{1}{1+tan^2a}\)
\(\Rightarrow cosa=-\frac{1}{\sqrt{1+tan^2a}}=-\frac{\sqrt{3}}{3}\); \(sina=-\sqrt{1-cos^2a}=-\frac{\sqrt{6}}{3}\)
\(cota=\frac{1}{tana}=\frac{\sqrt{2}}{2}\)
c/ \(sina=\sqrt{1-cos^2a}=\frac{\sqrt{5}}{5}\); \(tana=\frac{sina}{cosa}=\frac{1}{2}\); \(cota=\frac{1}{tana}=2\)
d/ \(sina=\sqrt{1-cos^2a}=\frac{\sqrt{209}}{15}\); \(tana=\frac{sina}{cosa}=\frac{\sqrt{209}}{4}\); \(cota=\frac{1}{tana}=\frac{4}{\sqrt{209}}\)
e/ \(\frac{1}{sin^2a}=1+cot^2a\Rightarrow sin^2a=\frac{1}{1+cot^2a}\Rightarrow sina=\frac{-1}{\sqrt{1+cot^2a}}\)
\(\Rightarrow sina=-\frac{\sqrt{10}}{10}\); \(cosa=\sqrt{1-sin^2a}=\frac{3\sqrt{10}}{10}\); \(cota=\frac{1}{tana}=-\frac{1}{3}\)
f/ \(cosa=-\frac{1}{\sqrt{1+tan^2a}}=-\frac{\sqrt{5}}{5}\); \(sina=tana.cosa=\frac{2\sqrt{5}}{5}\); \(cota=\frac{1}{tana}=-\frac{1}{2}\)
g/ Đề sai, trong khoảng \(\pi< a< \frac{3\pi}{2}\) thì \(\left\{{}\begin{matrix}sina< 0\\cosa< 0\end{matrix}\right.\) nên \(tana>0\)
\(\Rightarrow tana\) không thể nhận giá trị âm, ko có góc \(\alpha\)
\(\pi< a< \frac{3\pi}{2}\Rightarrow cosa< 0\)
\(\Rightarrow cosa=-\sqrt{1-sin^2a}=-0,8\)
\(sin2a=2sina.cosa=0,96\)
\(tana=\frac{sina}{cosa}=0,75\)
\(E=\frac{cosx}{sinx}+\frac{sinx}{1+cosx}=\frac{cosx+cos^2x+sin^2x}{sinx\left(1+cosx\right)}=\frac{cosx+1}{sinx\left(1+cosx\right)}=\frac{1}{sinx}\)
17.
\(\frac{\pi}{2}< a< \pi\Rightarrow cosa< 0\Rightarrow cosa=-\sqrt{1-sin^2a}=-\frac{12}{13}\)
\(0< b< \frac{\pi}{2}\Rightarrow sinb>0\Rightarrow sinb=\sqrt{1-cos^2b}=\frac{4}{5}\)
\(sin\left(a+b\right)=sina.cosb+cosa.sinb=\frac{5}{13}.\frac{3}{5}-\frac{12}{13}.\frac{4}{5}=-\frac{33}{65}\)
18.
\(K=sin\frac{2\pi}{7}+sin\frac{6\pi}{7}+sin\frac{4\pi}{7}\)
\(\Leftrightarrow K.sin\frac{\pi}{7}=sin\frac{\pi}{7}.sin\frac{2\pi}{7}+sin\frac{\pi}{7}.sin\frac{4\pi}{7}+sin\frac{\pi}{7}.sin\frac{6\pi}{7}\)
\(=\frac{1}{2}\left(cos\frac{\pi}{7}-cos\frac{3\pi}{7}+cos\frac{\pi}{7}-cos\frac{5\pi}{7}+cos\frac{5\pi}{7}-cos\frac{7\pi}{7}\right)\)
\(=\frac{1}{2}\left(cos\frac{\pi}{7}-cos\pi\right)=\frac{1}{2}\left(cos\frac{\pi}{7}+1\right)=\frac{1}{2}\left(2cos^2\frac{\pi}{14}-1+1\right)=cos^2\frac{\pi}{14}\)
\(\Leftrightarrow K.2.sin\frac{\pi}{14}.cos\frac{\pi}{14}=cos^2\frac{\pi}{14}\)
\(\Leftrightarrow2K=\frac{cos\frac{\pi}{14}}{sin\frac{\pi}{14}}=cot\frac{\pi}{14}=a\Rightarrow K=\frac{a}{2}\)
\(sina=\frac{3}{5}\Rightarrow sin^2a=\frac{9}{25}\) ; \(cos^2a=1-\frac{9}{25}=\frac{16}{25}\)
\(A=\frac{cota+tana}{cota-tana}=\frac{sina.cosa\left(cota+tana\right)}{sina.cosa\left(cota-tana\right)}=\frac{cos^2a+sin^2a}{cos^2a-sin^2a}=\frac{1}{cos^2a-sin^2a}=\frac{1}{\frac{16}{25}-\frac{9}{25}}=\frac{25}{7}\)
\(B=\frac{sin^2a-cos^2a}{sin^2a-3cos^2a}=\frac{\frac{sin^2a}{sin^2a}-\frac{cos^2a}{sin^2a}}{\frac{sin^2a}{sin^2a}-\frac{3cos^2a}{sin^2a}}=\frac{1-cot^2a}{1-3cot^2a}=\frac{1-\left(-\frac{1}{3}\right)^2}{1-3\left(-\frac{1}{3}\right)^2}=\)
\(C_1=sin^2a+cos^2a+cos^2a=1+cos^2a=1+\frac{1}{1+tan^2a}=1+\frac{1}{1+\left(-2\right)^2}\)
\(C_2=\left(sin^2a+cos^2a\right)\left(sin^2a-cos^2a\right)=sin^2a-cos^2a=1-2cos^2a\)
\(=1-\frac{2}{1+tan^2a}=1-\frac{2}{1+\left(-2\right)^2}\)
Do \(0< a< \frac{\pi}{2}\Rightarrow sina>0\)
\(sin^2a+cos^2a=1\Rightarrow sina=\sqrt{1-cos^2a}=\sqrt{1-\left(\frac{15}{17}\right)^2}=\frac{8}{17}\)
\(cos2a=2cos^2a-1=2.\left(\frac{15}{17}\right)^2-1=\frac{161}{289}\)
\(0< a< \frac{\pi}{2}\Rightarrow\left\{{}\begin{matrix}sina>0\\cosa>0\end{matrix}\right.\)
\(1+tan^2a=\frac{1}{cos^2a}\Rightarrow cos^2a=\frac{1}{1+tan^2a}\Rightarrow cosa=\frac{1}{\sqrt{1+tan^2a}}\)
\(\Rightarrow cosa=\frac{1}{2}\Rightarrow sina=cosa.tana=\frac{\sqrt{3}}{2}\)
\(cos2a=2cos^2a-1=-\frac{1}{2}\)
\(sin2a=2sina.cosa=\frac{\sqrt{3}}{2}\)
\(\Rightarrow sin\left(2a-\frac{\pi}{3}\right)=sin2a.cos\frac{\pi}{3}-cos2a.sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\)
\(tan\left(a+\frac{\pi}{4}\right)=\frac{tana+tan\frac{\pi}{4}}{1-tana.tan\frac{\pi}{4}}=-2-\sqrt{3}\)
Ta có \(0< a< \dfrac{\Pi}{2}\)
=>Điểm đầu và cuối của a thuộc góc phần tư thứ nhất
=> sin a > 0 và cos a >0
Có \(cos^2a+sin^2a=1\)
\(\Rightarrow\left(\dfrac{3}{5}\right)^2+sin^2a=1\)\(\Rightarrow sin^2a=\dfrac{16}{25}\)
\(\Rightarrow sina=\dfrac{4}{5}\)
\(sin2a=2sinacosa=2.\dfrac{4}{5}.\dfrac{3}{5}=\dfrac{24}{5}\)