e ms lớp 8

ai k minh giai cho

\(\sin3x-\cos5x=0\Leftrightarrow\cos5x=\sin3x\Leftrightarrow\cos5x=\cos\left(\frac{\pi}{2}-3x\right)\)

\(\Leftrightarrow5x=\pm\left(\frac{\pi}{2}-3x\right)+k2\pi\)

\(\Leftrightarrow\orbr{\begin{cases}x=\frac{\pi}{16}+k\frac{\pi}{4}\\x=-\frac{\pi}{4}+k\pi,k\in Z\end{cases}}\)

Đáp án C

Bài 7. a) sin 3x - cos 5x = 0 ⇔ cos 5x = sin 3x ⇔ cos 5x = cos ( - 3x) ⇔

b) tan 3x . tan x = 1 ⇔ . Điều kiện : cos 3x . cos x # 0.

Với điều kiện này phương trình tương đương với

cos 3x . cos x = sin 3x . sinx ⇔ cos 3x . cos x - sin 3x . sinx = 0 ⇔ cos 4x = 0.

Do đó

tan 3x . tan x = 1 ⇔

⇔ cos 2x = ⇔ cos 4x = 0

⇔

\( 2)\sin x + \sin 2x + \sin 3x = 0\\ \Leftrightarrow 2\sin 2x.\cos x + \sin 2x = 0\\ \Leftrightarrow \sin 2x\left( {2\cos x + 1} \right) = 0\\ \Leftrightarrow \left[ \begin{array}{l} \sin 2x = 0\\ 2\cos x + 1 = 0 \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} 2x = k\pi \\ \cos x = \dfrac{{ - 1}}{2} \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} x = \dfrac{{k\pi }}{2}\\ x = \pm \dfrac{{2\pi }}{3} + k2\pi \end{array} \right.\left( {k \in \mathbb{Z} } \right) \)

\( 3)\sin x + \sin 2x + \sin 3x + \sin 4x = 0\\ \Leftrightarrow \left( {\sin x + \sin 4x} \right) + \left( {\sin 2x + \sin 3x} \right) = 0\\ \Leftrightarrow 2\sin \dfrac{{5x}}{2}.\cos \dfrac{{3x}}{2} + 2\sin \dfrac{{5x}}{2}.\cos \dfrac{x}{2} = 0\\ \Leftrightarrow \sin \dfrac{{5x}}{2}.\left( {\cos \dfrac{{3x}}{2} + \cos \dfrac{x}{2}} \right) = 0\\ \Leftrightarrow \sin \dfrac{{5x}}{2}.2\cos x.\cos \dfrac{x}{2} = 0\\ \Leftrightarrow \left[ \begin{array}{l} \sin \dfrac{{5x}}{2} = 0\\ 2\cos x = 0\\ \cos \dfrac{x}{2} = 0 \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} x = \dfrac{{2k\pi }}{5}\\ x = \dfrac{\pi }{2} + k\pi \\ x = \pi + 2k\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right) \)

\(\Leftrightarrow\sqrt{3}cos5x-\left(sin5x+sinx\right)-sinx=0\)

\(\Leftrightarrow\sqrt{3}cos5x-sin5x=2sinx\)

\(\Leftrightarrow\frac{\sqrt{3}}{2}cos5x-\frac{1}{2}sin5x=sinx\)

\(\Leftrightarrow sin\left(\frac{\pi}{3}-5x\right)=sinx\)

\(\Leftrightarrow...\)

\(\Leftrightarrow cos5x=-sin3x\)

\(\Leftrightarrow cos5x=cos\left(\dfrac{\pi}{2}+3x\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}5x=\dfrac{\pi}{2}+3x+k2\pi\\5x=-\dfrac{\pi}{2}-3x+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+k\pi\\x=-\dfrac{\pi}{16}+\dfrac{k\pi}{4}\end{matrix}\right.\)

a)pt\(\Leftrightarrow cosx\left(cosx+1\right)+sinx.sin^2x=0\)

\(\Leftrightarrow cosx\left(cosx+1\right)+sinx\left(1-cos^2x\right)=0\)

\(\Leftrightarrow\left(cosx+1\right)\left(cosx+sinx-sinx.cosx\right)=0\)

\(\Leftrightarrow\left[\begin{array}{nghiempt}cosx=1\Leftrightarrow x=\pi+k2\pi\\cosx+sinx-sinx.cosx=0\left(\cdot\right)\end{array}\right.\)

Xét pt(*):

Đặt \(t=cosx+sinx,t\in\left[-\sqrt{2};\sqrt{2}\right]\Rightarrow sinx.cosx=\frac{t^2-1}{2}\)

(*) trở thành:\(t^2-2t-1=0\Leftrightarrow\left[\begin{array}{nghiempt}t=1-\sqrt{2}\\t=1+\sqrt{2}\left(L\right)\end{array}\right.\)

+)\(t=1-\sqrt{2}\Rightarrow\sqrt{2}sin\left(x+\frac{\pi}{4}\right)=1-\sqrt{2}\\ \Leftrightarrow\left[\begin{array}{nghiempt}x=-\frac{\pi}{4}+arcsin\left(\frac{-2+\sqrt{2}}{2}\right)+k2\pi\\x=-\frac{5\pi}{4}-arcsin\left(\frac{-2+\sqrt{2}}{2}\right)+k2\pi\end{cases}\left(k\in Z\right)}\)