Chương 1: HÀM SỐ LƯỢNG GIÁC. PHƯƠNG TRÌNH LƯỢNG GIÁC

2.

\(\Leftrightarrow sin\left(\dfrac{x}{2}+10^0\right)=-\dfrac{1}{2}\)

\(\Leftrightarrow sin\left(\dfrac{x}{2}+10^0\right)=sin\left(-30^0\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{x}{2}+10^0=-30^0+k360^0\\\dfrac{x}{2}+10^0=210^0+k360^0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{x}{2}=-40^0+k360^0\\\dfrac{x}{2}=200^0+k360^0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-80^0+k720^0\\x=400^0+k720^0\end{matrix}\right.\)

5.

\(\Leftrightarrow cot\left(2x+\dfrac{5\pi}{2}\right)=-\dfrac{\sqrt{3}}{3}\)

\(\Leftrightarrow cot\left(2x+\dfrac{\pi}{2}+2\pi\right)=-\dfrac{\sqrt[]{3}}{3}\)

\(\Leftrightarrow cot\left(2x+\dfrac{\pi}{2}\right)=-\dfrac{\sqrt{3}}{3}\)

\(\Leftrightarrow tan2x=\dfrac{\sqrt{3}}{3}\)

\(\Leftrightarrow2x=\dfrac{\pi}{6}+k\pi\)

\(\Leftrightarrow x=\dfrac{\pi}{12}+\dfrac{k\pi}{2}\)

1.

\(\Leftrightarrow cos\left(x-\dfrac{\pi}{6}\right)=-\dfrac{1}{2}\)

\(\Leftrightarrow cos\left(x-\dfrac{\pi}{6}\right)=cos\left(\dfrac{2\pi}{3}\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}x-\dfrac{\pi}{6}=\dfrac{2\pi}{3}+k2\pi\\x-\dfrac{\pi}{6}=-\dfrac{2\pi}{3}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{5\pi}{6}+k2\pi\\x=-\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\)

4.

\(\Leftrightarrow sin\left(x+\dfrac{\pi}{4}\right)=-\dfrac{\sqrt{2}}{2}\)

\(\Leftrightarrow sin\left(x+\dfrac{\pi}{4}\right)=sin\left(-\dfrac{\pi}{4}\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}x+\dfrac{\pi}{4}=-\dfrac{\pi}{4}+k2\pi\\x+\dfrac{\pi}{4}=\dfrac{5\pi}{4}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{2}+k2\pi\\x=\pi+k2\pi\end{matrix}\right.\)

1) \(2cos\left(x-\dfrac{\pi}{6}\right)=-1\Rightarrow cos\left(x-\dfrac{\pi}{6}\right)=-\dfrac{1}{2}\)

     \(\Rightarrow cos\left(x-\dfrac{\pi}{6}\right)=cos\dfrac{2\pi}{3}\)

     \(\Rightarrow\left[{}\begin{matrix}x-\dfrac{\pi}{6}=\dfrac{2\pi}{3}+k2\pi\\x-\dfrac{\pi}{6}=-\dfrac{2\pi}{3}+k2\pi\end{matrix}\right.\)  \(\Rightarrow\left[{}\begin{matrix}x=\dfrac{5\pi}{6}+k2\pi\\x=-\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\) \(\left(k\in Z\right)\)

2) \(2sin\left(x+\dfrac{\pi}{4}\right)+\sqrt{2}=0\)

\(\Rightarrow sin\left(x+\dfrac{\pi}{4}\right)=-\dfrac{\sqrt{2}}{2}\)

\(\Rightarrow sin\left(x+\dfrac{\pi}{4}\right)=sin\left(-\dfrac{\pi}{4}\right)\)

\(\Rightarrow\left[{}\begin{matrix}x+\dfrac{\pi}{4}=-\dfrac{\pi}{4}+k2\pi\\x+\dfrac{\pi}{4}=\pi+\dfrac{\pi}{4}+k2\pi\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{2}+k2\pi\\x=\pi+k2\pi\end{matrix}\right.\) \(\left(k\in Z\right)\)

Các câu 1,2,4,5 làm rồi

3.

\(2cos\left(x-\pi\right)-1=0\)

\(\Leftrightarrow cos\left(x-\pi\right)=\dfrac{1}{2}\)

\(\Leftrightarrow-cosx=\dfrac{1}{2}\)

\(\Leftrightarrow cosx=-\dfrac{1}{2}\)

\(\Leftrightarrow x=\pm\dfrac{2\pi}{3}+k2\pi\)

6.

\(\Leftrightarrow\left[{}\begin{matrix}cosx=-2< -1\left(vô-nghiệm\right)\\cos2x=\dfrac{1}{3}\end{matrix}\right.\)

\(\Leftrightarrow2x=\pm arccos\left(\dfrac{1}{3}\right)+k2\pi\)

\(\Leftrightarrow x=\pm\dfrac{1}{2}arccos\left(\dfrac{1}{3}\right)+k\pi\)

1.

\(sin2x=-\dfrac{1}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}2x=-\dfrac{\pi}{6}+k2\pi\\2x=\dfrac{7\pi}{6}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{12}+k\pi\\x=\dfrac{7\pi}{12}+k\pi\end{matrix}\right.\) 

\(\Rightarrow x=\left\{\dfrac{7\pi}{12};\dfrac{11\pi}{12}\right\}\)

3.

\(tan\left(2x-15^0\right)=1\)

\(\Rightarrow2x-15^0=45^0+k180^0\)

\(\Rightarrow x=30^0+k90^0\)

\(\Rightarrow x=\left\{-60^0;30^0\right\}\)

2.

\(cos\left(x-5\right)=\dfrac{\sqrt{3}}{2}\)

\(\Leftrightarrow cos\left(x-5\right)=cos\left(\dfrac{\pi}{6}\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}x-5=\dfrac{\pi}{6}+k2\pi\\x-5=-\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=5+\dfrac{\pi}{6}+k2\pi\\x=5-\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)

\(\Rightarrow x=\left\{5-\dfrac{11\pi}{6};5-\dfrac{13\pi}{6}\right\}\)

4.

\(\Leftrightarrow cot3x=cot\left(-\dfrac{\pi}{3}\right)\)

\(\Leftrightarrow3x=-\dfrac{\pi}{3}+k\pi\)

\(\Leftrightarrow x=-\dfrac{\pi}{9}+\dfrac{k\pi}{3}\)

\(-\dfrac{\pi}{2}< x< 0\Rightarrow-\dfrac{\pi}{2}< -\dfrac{\pi}{9}+\dfrac{k\pi}{3}< 0\)

\(\Rightarrow-\dfrac{7}{6}< k< \dfrac{1}{3}\)

\(\Rightarrow k=\left\{-1;0\right\}\)

\(\Rightarrow x=\left\{-\dfrac{4\pi}{9};-\dfrac{\pi}{9}\right\}\)

ĐK: \(x\ne\dfrac{\pi}{12}+\dfrac{k\pi}{2};x\ne\dfrac{\pi}{6}+\dfrac{k\pi}{3}\)

\(tan\left(2x+\dfrac{\pi}{3}\right)+tan3x=0\)

\(\Leftrightarrow\dfrac{sin\left(5x+\dfrac{\pi}{3}\right)}{cos\left(2x+\dfrac{\pi}{3}\right).cos3x}=0\)

\(\Leftrightarrow sin\left(5x+\dfrac{\pi}{3}\right)=0\)

\(\Leftrightarrow5x+\dfrac{\pi}{3}=k\pi\)

\(\Leftrightarrow x=-\dfrac{\pi}{15}+\dfrac{k\pi}{5}\)

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8.

\(cosx-2sin^2\dfrac{x}{2}=0\)

\(\Leftrightarrow cosx+1-2sin^2\dfrac{x}{2}-1=0\)

\(\Leftrightarrow2cosx-1=0\)

\(\Leftrightarrow cosx=\dfrac{1}{2}\)

\(\Leftrightarrow x=\pm\dfrac{\pi}{3}+k2\pi\)

9.

\(cos^4x-sin^4x=\dfrac{\sqrt{2}}{2}\)

\(\Leftrightarrow\left(cos^2x+sin^2x\right)\left(cos^2x-sin^2x\right)=\dfrac{\sqrt{2}}{2}\)

\(\Leftrightarrow cos2x=\dfrac{\sqrt{2}}{2}\)

\(\Leftrightarrow2x=\pm\dfrac{\pi}{4}+k2\pi\)

\(\Leftrightarrow x=\pm\dfrac{\pi}{8}+k\pi\)