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bài 1) ta có : \(G=cos\left(\alpha-5\pi\right)+sin\left(\dfrac{-3\pi}{2}+\alpha\right)-tan\left(\dfrac{\pi}{2}+\alpha\right).cot\left(\dfrac{3\pi}{2}-\alpha\right)\)
\(G=cos\left(\alpha-\pi\right)+sin\left(\dfrac{\pi}{2}+\alpha\right)-tan\left(\dfrac{\pi}{2}+\alpha\right).cot\left(\dfrac{\pi}{2}-\alpha\right)\)
\(G=cos\left(\pi-\alpha\right)+sin\left(\dfrac{\pi}{2}-\left(-\alpha\right)\right)-tan\left(\pi+\alpha-\dfrac{\pi}{2}\right).cot\left(\dfrac{\pi}{2}-\alpha\right)\) \(G=cos\left(\alpha\right)+cos\left(\alpha\right)+tan\left(\dfrac{\pi}{2}-\alpha\right).cot\left(\dfrac{\pi}{2}-\alpha\right)=2cos\alpha+1\) bài 2) ta có : \(H=cot\left(\alpha\right).cos\left(\alpha+\dfrac{\pi}{2}\right)+cos\left(\alpha\right)-2sin\left(\alpha-\pi\right)\) \(H=cot\left(\alpha\right).cos\left(\dfrac{\pi}{2}-\left(-\alpha\right)\right)+cos\left(\alpha\right)+2sin\left(\pi-\alpha\right)\) \(H=-cot\left(\alpha\right).sin\left(\alpha\right)+cos\left(\alpha\right)+2sin\left(\alpha\right)\)
\(H=-cos\alpha+cos\alpha+2sin\alpha=2sin\alpha\)
\(sin\alpha=\dfrac{3}{4}\)
\(sin^2\alpha+cos^2\alpha=1\)
\(\Leftrightarrow cos^2\alpha=1-sin^2\alpha\)
\(\Leftrightarrow cos^2\alpha=1-\dfrac{9}{16}=\dfrac{7}{16}\)
\(\Leftrightarrow cos\alpha=-\dfrac{\sqrt[]{7}}{4}\left(\dfrac{\pi}{2}< \alpha< \pi\right)\)
\(tan\alpha=\dfrac{sin\alpha}{cos\alpha}=\dfrac{\dfrac{3}{4}}{-\dfrac{\sqrt[]{7}}{4}}=-\dfrac{3}{\sqrt[]{7}}=-\dfrac{3\sqrt[]{7}}{7}\)
\(\Rightarrow cot\alpha=\dfrac{1}{tan\alpha}=-\dfrac{\sqrt[]{7}}{3}\)
Lời giải:
Theo công thức lượng giác:
\(F=\sin (\pi +a)-\cos (\frac{\pi}{2}-a)+\cot (2\pi -a)+\tan (\frac{3\pi}{2}-a)\)
\(=-\sin a-\sin a+\cot (\pi -a)+\tan (\frac{\pi}{2}-a)\)
\(=-2\sin a-\cot a+\cot a=-2\sin a\)
a)\(sin\left(\alpha+\dfrac{\pi}{2}\right)=cos\left[\dfrac{\pi}{2}-\left(\alpha+\dfrac{\pi}{2}\right)\right]=cos\left(-\alpha\right)=cos\alpha\).
b) \(cos\left(x+\dfrac{\pi}{2}\right)=sin\left[\dfrac{\pi}{2}-\left(x+\dfrac{\pi}{2}\right)\right]=sin\left(-x\right)=-sinx\).
c) \(tan\left(\alpha+\dfrac{\pi}{2}\right)=\dfrac{sin\left(\alpha+\dfrac{\pi}{2}\right)}{cos\left(\alpha+\dfrac{\pi}{2}\right)}=\dfrac{cos\alpha}{-sin\alpha}=-cot\alpha\).
d) \(cot\left(\alpha+\dfrac{\pi}{2}\right)=\dfrac{cos\left(\alpha+\dfrac{\pi}{2}\right)}{sin\left(\alpha+\dfrac{\pi}{2}\right)}=\dfrac{-sin\alpha}{cos\alpha}=-tan\alpha\).
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 -_-
G = \(cos\left(a+\pi-6\text{}\text{}\pi\right)+sin\left(-2\pi+\dfrac{\pi}{2}+a\right)-tan\left(\dfrac{\pi}{2}+a\right)\cdot cot\left(\pi+\dfrac{\pi}{2}-a\right)\)
= \(cos\left(a+\pi\right)+sin\left(\dfrac{\pi}{2}+a\right)-tan\left(\dfrac{\pi}{2}+a\right)\cdot cot\left(\dfrac{\pi}{2}-a\right)\)
= \(-cosa+cosa-\left(-cota\cdot tana\right)=1\)
Lời giải:
$\frac{\pi}{2}< a< \pi$ nên $\sin a>0; \cos a< 0$
$-3=\tan a=\frac{\sin a}{\cos a}\Rightarrow \sin a=-3\cos a$
$\Rightarrow \sin ^2a=9\cos ^2a$
$\Rightarrow 10\sin ^2a=9(\sin ^2a+\cos ^2a)=9$
$\Rightarrow \sin ^2a=\frac{9}{10}$
$\Rightarrow \sin a=\frac{3}{\sqrt{10}}$
$\cos a=\frac{\sin a}{-3}=\frac{-1}{\sqrt{10}}$
$\cot a=\frac{1}{\tan a}=\frac{-1}{3}$
\(VT=\dfrac{-tan\left(\dfrac{\pi}{2}-a\right)cos\left(2\pi-\dfrac{\pi}{2}+a\right)-sin^3\left(4\pi-\dfrac{\pi}{2}-a\right)}{cos\left(\dfrac{\pi}{2}-a\right)tan\left(2\pi-\dfrac{\pi}{2}+a\right)}\)
\(=\dfrac{-cota.sina+sin^3\left(\dfrac{\pi}{2}+a\right)}{sina.\left(-cota\right)}=\dfrac{-cosa+cos^3a}{-cosa}=1-cos^2a=sin^2a\)
Do \(\pi< a< \frac{3\pi}{2}\Rightarrow cosa< 0\)
\(cosa=-\sqrt{1-sin^2a}=-\sqrt{1-0,6^2}=-\frac{4}{5}\)
\(tana=\frac{sina}{cosa}=-\frac{3}{4}\)
\(cota=\frac{1}{tana}=-\frac{4}{3}\)
\(0< a< \frac{\pi}{2}\Rightarrow sina;cosa;tana>0\)
\(tana+\frac{1}{tana}=3\Leftrightarrow tan^2a-3tana+1=0\) \(\Rightarrow\left[{}\begin{matrix}tana=\frac{3-\sqrt{5}}{2}\\tana=\frac{3+\sqrt{5}}{2}\end{matrix}\right.\)
- Với \(tana=\frac{3-\sqrt{5}}{2}\)
\(\Rightarrow cota=\frac{1}{tana}=\frac{3+\sqrt{5}}{2}\)
\(1+tan^2a=\frac{1}{cos^2a}\Rightarrow cosa=\frac{1}{\sqrt{1+tan^2a}}=\frac{2}{\sqrt{18-6\sqrt{5}}}\)
\(sina=\sqrt{1-cos^2a}=\frac{2}{\sqrt{18+6\sqrt{5}}}\)
\(cos\left(\frac{3\pi}{2}-a\right)=cos\left(2\pi-\frac{\pi}{2}-a\right)=-sina=...\)
\(sin\left(2\pi+a\right)=sina=...\)
\(tan\left(\pi-a\right)=-tana=...\)
\(cot\left(\pi+a\right)=cota=...\)
TH2: \(tana=\frac{3+\sqrt{5}}{2}\)
Tương tự như trên