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Ta có: `sin^2 \alpha +cos^2 \alpha=1`
`<=>9/25+cos^2 \alpha=1`
`<=>cos^2 \alpha =16/25`
`=>cos \alpha =[+-4]/5`
Lại có: `1+tan^2 \alpha =1/[cos^2 \alpha]`
`<=>1+tan^2 \alpha=1/[16/25]=>tan^2 \alpha=9/16`
`@` Với `cos \alpha =4/5=>A=[2.(3/5)^2+3]/[2. 4/5-9/16]=1488/415`
`@` Với `cos \alpha =-4/5=>A=[2.(3/5)^2+3]/[2. [-4]/5-9/16]=-1488/865`
Lời giải:
$\cos a=\sqrt{1-\sin ^2a}=\frac{4}{5}$
$\tan a=\frac{\sin a}{\cos a}=\frac{3}{5}: \frac{4}{5}=\frac{3}{4}$
$A=2\tan a+\cos a=2.\frac{3}{4}+\frac{4}{5}=\frac{23}{10}$
Ta có :\(sin^2a+cos^2a=1\)
Thay số: \(\left(\frac{2}{3}\right)^2\)\(+cos^2a=1\)\(\Rightarrow cos^2a=\frac{5}{9}\)
A=\(2sin^2a+5cos^2a\)\(\Rightarrow2.\frac{4}{9}+5.\frac{5}{9}\)\(\Rightarrow A=\frac{11}{3}\)
Ta có: \(cot\alpha=\dfrac{cos\alpha}{sin\alpha}=\dfrac{cos^2\alpha}{sin\alpha.cos\alpha}=\sqrt{5}\)
Lại có: \(\dfrac{1}{cot\alpha}=tan\alpha=\dfrac{sin\alpha}{cos\alpha}=\dfrac{sin^2\alpha}{cos\alpha.sin\alpha}=\dfrac{1}{\sqrt{5}}\)
\(\Rightarrow A=\dfrac{cos^2\alpha}{sin\alpha.cos\alpha}+\dfrac{sin^2\alpha}{sin\alpha.cos\alpha}=\sqrt{5}+\dfrac{1}{\sqrt{5}}=\dfrac{6}{\sqrt{5}}=\dfrac{6\sqrt{5}}{5}\)
Ta có : cot α = \(\sqrt{5}\Rightarrow\dfrac{cos\alpha}{sin\alpha}=\sqrt{5}\Rightarrow cos\alpha=\sqrt{5}.sin\alpha\)
\(A=\dfrac{sin^2\alpha+cos^2\alpha}{sin\alpha.cos\alpha}\)
\(A=\dfrac{sin^2\alpha+\left(\sqrt{5}sin\alpha\right)^2}{sin\alpha.\sqrt{5}sin\alpha}=\dfrac{sin^2\alpha+5sin^2\alpha}{\sqrt{5}sin^2\alpha}\)
\(A=\dfrac{6sin^2\alpha}{\sqrt{5}sin^2\alpha}=\dfrac{6}{\sqrt{5}}=\dfrac{6\sqrt{5}}{5}\)
\(\hept{\begin{cases}sin^2a+c\text{os}^2a=1\\sina=2cosa\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}sina=\frac{2}{\sqrt{5}}\\c\text{os}a=\frac{1}{\sqrt{5}}\end{cases}}\)hoặc \(\orbr{\begin{cases}sina=-\frac{2}{\sqrt{5}}\\c\text{os}a=-\frac{1}{\sqrt{5}}\end{cases}}\)
Thế vô đi
\(A=\sin^6\alpha+\cos^6\alpha+3.1.\sin^2\alpha.\cos^2\alpha=\left(\sin^2\alpha\right)^3+\left(\cos^2\alpha\right)^3+3.\sin^2\alpha.\cos^2\alpha\left(\sin^2\alpha+\cos^2\alpha\right)\)
\(=\left(\sin^2\alpha+\cos^2\alpha\right)^2=1^2=1\)
\(A=2\left(sin^2a+cos^2a\right)+3cos^2a=2+3\cdot cos^2a\)
mặt khác: \(sina=\dfrac{2}{3}\Leftrightarrow a=sin^{-1}\left(\dfrac{2}{3}\right)\)
thay vào A , ta được:
\(A=2+3\cdot sin^{-1}\left(\dfrac{2}{3}\right)=....\) (số xấu quá!)
A=2(sin2a + cos2a) +3 cos2a=2+ 3 cos2a
ta có sin2a+cos2a=1
(2/3)2 + cos2a =1
cosa=\(\dfrac{\sqrt{5}}{3}\)
A=....