Cho cos a= \(\frac{-3}{5}\). Hãy tính sin a, cos a, tan a
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![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
vì \(\sin\alpha\ne0\)nên chia cả tử và mẫu của biểu thức cho \(\sin\alpha\),ta được :
\(\frac{\cos\alpha+\sin\alpha}{\cos\alpha-\sin\alpha}=\frac{\frac{\cos\alpha}{\sin\alpha}+1}{\frac{\cos\alpha}{\sin\alpha}-1}=\frac{\cot\alpha+1}{\cot\alpha-1}=\frac{\frac{1}{\tan\alpha}+1}{\frac{1}{\tan\alpha}-1}=\frac{2+1}{2-1}=3\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(tana-5cota+4=0\Rightarrow tana-\dfrac{5}{tana}+4=0\)
\(\Rightarrow tan^2a+4tana-5=0\Rightarrow\left[{}\begin{matrix}tana=1\\tana=-5\end{matrix}\right.\)
\(A=\dfrac{4sina+2cosa}{3sina-cosa}=\dfrac{\dfrac{4sina}{cosa}+\dfrac{2cosa}{cosa}}{\dfrac{3sina}{cosa}-\dfrac{cosa}{cosa}}=\dfrac{4tana+2}{3tana-1}=\left[{}\begin{matrix}3\\\dfrac{9}{8}\end{matrix}\right.\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có:
\({\cos ^2}a + {\sin ^2}a = 1 \Rightarrow \sin a = \pm \frac{4}{5}\)
Do \(0 < a < \frac{\pi }{2} \Leftrightarrow \sin a = \frac{4}{5}\)
\(\tan a = \frac{{\sin a}}{{\cos a}} = \frac{4}{3}\)
Ta có;
\(\begin{array}{l}\sin \left( {a + \frac{\pi }{6}} \right) = \sin a.\cos \frac{\pi }{6} + \cos a.\sin \frac{\pi }{6} = \frac{4}{5}.\frac{{\sqrt 3 }}{2} + \frac{3}{5}.\frac{1}{2} = \frac{{3 + 4\sqrt 3 }}{{10}}\\\cos \left( {a - \frac{\pi }{3}} \right) = \cos a.\cos \frac{\pi }{3} + \sin a.\sin \frac{\pi }{3} = \frac{3}{5}.\frac{1}{2} + \frac{4}{5}.\frac{{\sqrt 3 }}{2} = \frac{{3 + 4\sqrt 3 }}{{10}}\\\tan \left( {a + \frac{\pi }{4}} \right) = \frac{{\tan a + \tan \frac{\pi }{4}}}{{1 - \tan a.tan\frac{\pi }{4}}} = \frac{{\frac{4}{3} + 1}}{{1 - \frac{4}{3}}} = - 7\end{array}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\begin{array}{l}\cos 2a = \frac{1}{3} \Leftrightarrow {\cos ^2}a - {\sin ^2}a = \frac{1}{3}\,\,\left( 1 \right)\\{\cos ^2}a + {\sin ^2}a = 1\,\,\,\,\left( 2 \right)\end{array}\)
Từ (1) và (2) \( \Rightarrow \left\{ \begin{array}{l}{\cos ^2}a = \frac{2}{3}\\{\sin ^2}a = \frac{1}{3}\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}\cos a = \pm \frac{{\sqrt 6 }}{3}\\\sin a = \pm \frac{{\sqrt 3 }}{3}\end{array} \right.\)
Do \(\frac{\pi }{2} < a < \pi \)\( \Rightarrow \left\{ \begin{array}{l}\cos a = \frac{{-\sqrt 6 }}{3}\\\sin a = \ \frac{{\sqrt 3 }}{3}\end{array} \right.\)
\(\Rightarrow \tan a = \frac{{\sin a}}{{\cos a}} = - \frac{{\sqrt 2 }}{2}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a: sin a=2/3
=>cos^2a=1-(2/3)^2=5/9
=>\(cosa=\dfrac{\sqrt{5}}{3}\)
\(tana=\dfrac{2}{3}:\dfrac{\sqrt{5}}{3}=\dfrac{2}{\sqrt{5}}\)
\(cota=1:\dfrac{2}{\sqrt{5}}=\dfrac{\sqrt{5}}{2}\)
b: cos a=1/5
=>sin^2a=1-(1/5)^2=24/25
=>\(sina=\dfrac{2\sqrt{6}}{5}\)
\(tana=\dfrac{2\sqrt{6}}{5}:\dfrac{1}{5}=2\sqrt{6}\)
\(cota=\dfrac{1}{2\sqrt{6}}=\dfrac{\sqrt{6}}{12}\)
c: cot a=1/tana=1/2
\(1+tan^2a=\dfrac{1}{cos^2a}\)
=>1/cos^2a=1+4=5
=>cos^2a=1/5
=>cosa=1/căn 5
\(sina=\sqrt{1-cos^2a}=\dfrac{2}{\sqrt{5}}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(B=\frac{2cosa-sina}{cosa+2sina}=\frac{2-tana}{1+2tana}=\frac{2-2+\sqrt{3}}{1+2\left(2-\sqrt{3}\right)}=\frac{\sqrt{3}}{5-2\sqrt{3}}\)
PS: Mấy cái như điều kiện xác định thì bạn tự làm nhé.
![](https://rs.olm.vn/images/avt/0.png?1311)
Tự chứng minh từng cái này rồi suy ra cái đó nhé b.
Ta có: \(sin\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}-sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}=sin^2\frac{A}{2}\)
Tương tự ta suy ra:
\(sin\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}+cos\frac{A}{2}sin\frac{B}{2}cos\frac{C}{2}+cos\frac{A}{2}cos\frac{B}{2}sin\frac{C}{2}=sin^2\frac{A}{2}+sin^2\frac{B}{2}+sin^2\frac{C}{2}+3sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}\left(1\right)\)
Tiếp theo chứng minh:
\(2sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}=\frac{cosA+cosB+cosC-1}{2}\left(2\right)\)
\(sin^2\frac{A}{2}+sin^2\frac{B}{2}+sin^2\frac{C}{2}=\frac{3}{2}-\frac{cosA+cosB+cosC}{2}\left(3\right)\)
\(tan\frac{A}{2}tan\frac{B}{2}+tan\frac{B}{2}tan\frac{C}{2}+tan\frac{C}{2}tan\frac{A}{2}=1\left(4\right)\)
Từ (1), (2), (3), (4) suy được điều phải chứng minh
![](https://rs.olm.vn/images/avt/0.png?1311)
1:
a: sin a=căn 3/2
\(cosa=\sqrt{1-sin^2a}=\sqrt{1-\dfrac{3}{4}}=\sqrt{\dfrac{1}{4}}=\dfrac{1}{2}\)
\(tana=\dfrac{\sqrt{3}}{2}:\dfrac{1}{2}=\sqrt{3}\)
cot a=1/tan a=1/căn 3
b: \(tana=2\)
=>cot a=1/tan a=1/2
\(1+tan^2a=\dfrac{1}{cos^2a}\)
=>\(\dfrac{1}{cos^2a}=5\)
=>cos^2a=1/5
=>cosa=1/căn 5
\(sina=\sqrt{1-cos^2a}=\sqrt{\dfrac{4}{5}}=\dfrac{2}{\sqrt{5}}\)
c: \(cosa=\sqrt{1-\left(\dfrac{5}{13}\right)^2}=\dfrac{12}{13}\)
tan a=5/13:12/13=5/12
cot a=1:5/12=12/5
![](https://rs.olm.vn/images/avt/0.png?1311)
a) Vì \(\frac{\pi }{2} < a < \pi \) nên \(\cos a < 0\)
Ta có: \({\sin ^2}a + {\cos ^2}a = 1\)
\(\Leftrightarrow \frac{1}{9} + {\cos ^2}a = 1\)
\(\Leftrightarrow {\cos ^2}a = 1 - \frac{1}{9}= \frac{8}{9}\)
\(\Leftrightarrow \cos a =\pm\sqrt { \frac{8}{9}} = \pm \frac{{2\sqrt 2 }}{3}\)
Vì \(\cos a < 0\) nên \(cos a =-\frac{{2\sqrt 2 }}{3}\)
Suy ra \(\tan a = \frac{{\sin a}}{{\cos a}} = \frac{{\frac{1}{3}}}{{ - \frac{{2\sqrt 2 }}{3}}} = - \frac{{\sqrt 2 }}{4}\)
Ta có: \(\sin 2a = 2\sin a\cos a = 2.\frac{1}{3}.\left( { - \frac{{2\sqrt 2 }}{3}} \right) = - \frac{{4\sqrt 2 }}{9}\)
\(\cos 2a = 1 - 2{\sin ^2}a = 1 - \frac{2}{9} = \frac{7}{9}\)
\(\tan 2a = \frac{{2\tan a}}{{1 - {{\tan }^2}a}} = \frac{{2.\left( { - \frac{{\sqrt 2 }}{4}} \right)}}{{1 - {{\left( { - \frac{{\sqrt 2 }}{4}} \right)}^2}}} = - \frac{{4\sqrt 2 }}{7}\)
b) Vì \(\frac{\pi }{2} < a < \frac{{3\pi }}{4}\) nên \(\sin a > 0,\cos a < 0\)
\({\left( {\sin a + \cos a} \right)^2} = {\sin ^2}a + {\cos ^2}a + 2\sin a\cos a = 1 + 2\sin a\cos a = \frac{1}{4}\)
Suy ra \(\sin 2a = 2\sin a\cos a = \frac{1}{4} - 1 = - \frac{3}{4}\)
Ta có: \({\sin ^2}a + {\cos ^2}a = 1\;\)
\( \Leftrightarrow \left( {\frac{1}{2} - {\cos }a} \right)^2 + {\cos ^2}a - 1 = 0\)
\( \Leftrightarrow \frac{1}{4} - \cos a + {\cos ^2}a + {\cos ^2}a - 1 = 0\)
\( \Leftrightarrow 2{\cos ^2}a - \cos a - \frac{3}{4} = 0\)
\( \Rightarrow \cos a = \frac{{1 - \sqrt 7 }}{4}\) (Vì \(\cos a < 0)\)
\(\cos 2a = 2{\cos ^2}a - 1 = 2.{\left( {\frac{{1 - \sqrt 7 }}{4}} \right)^2} - 1 = - \frac{{\sqrt 7 }}{4}\)
\(\tan 2a = \frac{{\sin 2a}}{{\cos 2a}} = \frac{{ - \frac{3}{4}}}{{ - \frac{{\sqrt 7 }}{4}}} = \frac{{3\sqrt 7 }}{7}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a, ta có \(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}\)
\(\frac{1}{3}\)= \(\frac{\sin\alpha}{\cos\alpha}\)
\(\cos\alpha\)= 3 \(\sin\alpha\)
ta có \(\frac{\cos\alpha+\sin\alpha}{\cos\alpha-\sin\alpha}\)= \(\frac{3\sin\alpha+\sin\alpha}{3\sin\alpha-\sin\alpha}\)= \(\frac{4\sin\alpha}{2\sin\alpha}\)= \(2\)
#mã mã#