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\(cot\left(a+\frac{\pi}{3}\right)=-\sqrt{3}\Leftrightarrow\frac{cos\left(a+\frac{\pi}{3}\right)}{sin\left(a+\frac{\pi}{3}\right)}=-\sqrt{3}\)
\(\Leftrightarrow cos\left(a+\frac{\pi}{3}\right)=-\sqrt{3}sin\left(a+\frac{\pi}{3}\right)\)
\(\Leftrightarrow cosa.cos\frac{\pi}{3}-sina.sin\frac{\pi}{3}=-\sqrt{3}\left(sinacos\frac{\pi}{3}+cosa.sin\frac{\pi}{3}\right)\)
\(\Leftrightarrow\frac{1}{2}cosa-\frac{\sqrt{3}}{2}sina+\frac{\sqrt{3}}{2}sina+\frac{3}{2}cosa=0\)
\(\Leftrightarrow2cosa=0\Rightarrow cosa=0\Rightarrow a=\frac{3\pi}{2}\)
\(\Rightarrow P=sin\left(\frac{3\pi}{2}+\frac{\pi}{6}\right)+cos\frac{3\pi}{2}=-\frac{\sqrt{3}}{2}\)
Câu 1:
\(tan\left(a+\frac{\pi}{4}\right)=1\Rightarrow a+\frac{\pi}{4}=\frac{\pi}{4}+k\pi\Rightarrow a=k\pi\) (\(k\in Z\) )
Do \(\frac{\pi}{2}< a< 2\pi\Rightarrow\frac{\pi}{2}< k\pi< 2\pi\Rightarrow\frac{1}{2}< k< 2\Rightarrow k=1\Rightarrow a=\pi\)
\(\Rightarrow P=cos\left(\pi-\frac{\pi}{6}\right)+sin\pi=-\frac{\sqrt{3}}{2}\)
Câu 2:
\(cot\left(a+\frac{\pi}{3}\right)=-\sqrt{3}=cot\left(-\frac{\pi}{6}\right)\)
\(\Rightarrow a+\frac{\pi}{3}=-\frac{\pi}{6}+k\pi\Rightarrow a=-\frac{\pi}{2}+k\pi\) (\(k\in Z\))
\(\Rightarrow\frac{\pi}{2}< -\frac{\pi}{2}+k\pi< 2\pi\Rightarrow-\pi< k\pi< \frac{5\pi}{2}\)
\(\Rightarrow-1< k< \frac{5}{2}\Rightarrow k=\left\{0;1;2\right\}\Rightarrow a=\left\{-\frac{\pi}{2};\frac{\pi}{2};\frac{3\pi}{2}\right\}\) \(\Rightarrow cosa=0\)
\(\Rightarrow P=sin\left(\pi+\frac{\pi}{6}\right)+0=-sin\frac{\pi}{6}=-\frac{1}{2}\)
Vậy đáp án sai
Bạn thay thử \(a=\frac{3\pi}{2}\) vào biểu thức ban đầu coi có đúng \(cot\left(a+\frac{\pi}{3}\right)=-\sqrt{3}\) ko là biết đáp án đúng hay sai liền mà
\(\pi< a< \frac{3\pi}{2}\Rightarrow\left\{{}\begin{matrix}sina< 0\\cosa< 0\end{matrix}\right.\)
\(\Rightarrow cosa=-\sqrt{1-sin^2a}=-\frac{4}{5}\) \(\Rightarrow tana=\frac{sina}{cosa}=\frac{3}{4}\)
b/ \(sina=\frac{\sqrt{3}}{3}???cosa=\frac{\sqrt{3}}{3}???\)
\(a\in\left(\frac{\pi}{2};\pi\right)\Rightarrow cosa< 0\Rightarrow cosa=-\sqrt{1-sin^2a}=-\frac{4}{5}\)
\(A=\frac{sin\left(4\pi-\frac{\pi}{2}-a\right)}{sin\left(a+\frac{\pi}{4}\right)-cosa}=\frac{-sin\left(a+\frac{\pi}{2}\right)}{sin\left(a+\frac{\pi}{4}\right)-cosa}=\frac{-cosa}{sina.cos\frac{\pi}{4}+cosa.sin\frac{\pi}{4}-cosa}\)
\(=\frac{-\frac{4}{5}}{\frac{3}{5}.\frac{\sqrt{2}}{2}-\frac{4}{5}.\frac{\sqrt{2}}{2}-\frac{4}{5}}=...\)
\(\left(sina-cosa\right)^2=2\Leftrightarrow sin^2a+cos^2a-2sina.cosa=2\)
\(\Leftrightarrow1-sin2a=2\Rightarrow sin2a=-1\)
\(\left(sina+cosa\right)^2=2\Leftrightarrow sin^2a+cos^2a+2sina.cosa=2\)
\(\Leftrightarrow1+sin2a=2\Rightarrow sin2a=1\)
\(\frac{3\pi}{2}< a< 2\pi\Rightarrow cosa>0\Rightarrow cosa=\sqrt{1-sin^2a}=\frac{1}{2}\)
\(\Rightarrow cos\left(a+\frac{\pi}{3}\right)=cosa.cos\frac{\pi}{3}-sina.sin\frac{\pi}{3}\)
\(=\frac{1}{2}.\frac{1}{2}-\left(-\frac{\sqrt{3}}{2}\right).\left(\frac{\sqrt{3}}{2}\right)=...\)
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 : \(H=cot\left(\alpha\right).cos\left(\alpha+\dfrac{\pi}{2}\right)+cos\left(\alpha\right)-2sin\left(\alpha-\pi\right)\) \(H=cot\left(\alpha\right).cos\left(\dfrac{\pi}{2}-\left(-\alpha\right)\right)+cos\left(\alpha\right)+2sin\left(\pi-\alpha\right)\) \(H=-cot\left(\alpha\right).sin\left(\alpha\right)+cos\left(\alpha\right)+2sin\left(\alpha\right)\)
\(H=-cos\alpha+cos\alpha+2sin\alpha=2sin\alpha\)
\(0< a< \frac{\pi}{2}\Rightarrow sina;cosa;tana>0\)
\(tana+\frac{1}{tana}=3\Leftrightarrow tan^2a-3tana+1=0\) \(\Rightarrow\left[{}\begin{matrix}tana=\frac{3-\sqrt{5}}{2}\\tana=\frac{3+\sqrt{5}}{2}\end{matrix}\right.\)
- 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