Giải bài 1.43, 1.44, 1.45 trang 40 - SBT Đại số và Giải tích lớp 11
Giải các phương trình sau (1.43 - 1.45)
1.43. \(\sin^2 x-\cos ^2 x=\cos 4x\)
1.44. \(\cos 3x-\cos 5x=\sin x\)
1.45. \(3\sin^2 x+4\cos x-2=0\)
1.43.
\(\begin{aligned} & {{\sin }^{2}}x-{{\cos }^{2}}x=\cos 4x \\ & \Leftrightarrow -\cos 2x=\cos 4x \\ & \Leftrightarrow \cos 4x+\cos 2x=0 \\ & \Leftrightarrow 2\cos 3x\cos x=0 \\ & \Leftrightarrow \left[ \begin{aligned} & \cos 3x=0 \\ & \cos x=0 \\ \end{aligned} \right. \\ & \Leftrightarrow \left[ \begin{aligned} & 3x=\dfrac{\pi }{2}+k\pi \\ & x=\dfrac{\pi }{2}+k\pi \\ \end{aligned} \right. \\ & \Leftrightarrow \left[ \begin{aligned} & x=\dfrac{\pi }{6}+k\dfrac{\pi }{3} \\ & x=\dfrac{\pi }{2}+k\pi \\ \end{aligned} \right.\\ &\Leftrightarrow x=\dfrac{\pi}{6}+k\dfrac{\pi}{3}\,\,\left( k\in \mathbb{Z} \right) \\ \end{aligned} \)
1. 44.
\(\begin{aligned} & \cos 3x-\cos 5x=\sin x \\ & \Leftrightarrow -2\sin 4x\sin \left( -x \right)=\sin x \\ & \Leftrightarrow 2\sin 4x\sin x-\sin x=0 \\ & \Leftrightarrow \sin x\left( 2\sin 4x-1 \right)=0 \\ & \Leftrightarrow \left[ \begin{aligned} & \sin x=0 \\ & \sin 4x=\dfrac{1}{2} \\ \end{aligned} \right. \\ & \Leftrightarrow \left[ \begin{aligned} & x=k\pi \\ & 4x=\dfrac{\pi }{6}+k2\pi \\ & 4x=\dfrac{5\pi }{6}+k2\pi \\ \end{aligned} \right. \\ & \Leftrightarrow \left[ \begin{aligned} & x=k\pi \\ & x=\dfrac{\pi }{24}+k\dfrac{\pi }{2} \\ & x=\dfrac{5\pi }{24}+k\dfrac{\pi }{2} \\ \end{aligned} \right.\,\,\,\left( k\in \mathbb{Z} \right) \\ \end{aligned} \)
1.45.
\(\begin{aligned} & 3{{\sin }^{2}}x+4\cos x-2=0 \\ & \Leftrightarrow 3-3{{\cos }^{2}}x+4\cos x-2=0 \\ & \Leftrightarrow 3{{\cos }^{2}}x-4\cos x-1=0 \\ & \Leftrightarrow \left[ \begin{aligned} & \cos x=\dfrac{2+\sqrt{7}}{3}\,\,\left( \text{loại} \right) \\ & \cos x=\dfrac{2-\sqrt{7}}{3} \\ \end{aligned} \right. \\ & \Leftrightarrow x=\pm \arccos \left( \dfrac{2-\sqrt{7}}{3} \right)+k2\pi ,\,\,k\in \mathbb{Z} \\ \end{aligned} \)