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

\(2sin^2x+\sqrt{3}sinx-3=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=\dfrac{\sqrt{3}}{2}\\sinx=-\sqrt{3}\left(loại\right)\end{matrix}\right.\)

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

a.

\(sin5x+sin3x+sin8x=0\)

\(\Leftrightarrow2sin4x.cosx+2sin4x.cos4x=0\)

\(\Leftrightarrow2sin4x\left(cosx+cos4x\right)=0\)

\(\Leftrightarrow4sin4x.cos\dfrac{5x}{2}cos\dfrac{3x}{2}=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sin4x=0\\cos\dfrac{5x}{2}=0\\cos\dfrac{3x}{2}=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}4x=k\pi\\\dfrac{5x}{2}=\dfrac{\pi}{2}+k\pi\\\dfrac{3x}{2}=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{k\pi}{4}\\x=\dfrac{\pi}{5}+\dfrac{k2\pi}{5}\\x=\dfrac{\pi}{3}+\dfrac{k2\pi}{3}\end{matrix}\right.\)

b.

\(\Leftrightarrow4cos^3x+6\sqrt{2}sinx.cosx=8cosx\)

\(\Leftrightarrow2cosx\left(2cos^2x+3\sqrt{2}sinx-4\right)=0\)

\(\Leftrightarrow cosx\left(-2sin^2x+3\sqrt{3}sinx-2\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cosx=0\\sinx=\sqrt{2}\left(loại\right)\\sinx=\dfrac{\sqrt{2}}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\x=\dfrac{\pi}{4}+k2\pi\\x=\dfrac{3\pi}{4}+k2\pi\end{matrix}\right.\)

\(\sqrt{2}sinx=2sin2x.cos2x-cos2x\)

\(\Leftrightarrow\sqrt{2}sinx=sin4x-cos4x\)

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

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

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

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{12}+\dfrac{k\pi}{3}\\x=\dfrac{\pi}{4}+\dfrac{k2\pi}{5}\end{matrix}\right.\)