sin3x - cos5x = 0

Vậy phương trình có hai họ nghiệm  (k ∈ Z).

1.

\(sinx-\sqrt{2}cos3x=\sqrt{3}cosx+\sqrt{2}sin3x\)

\(\Leftrightarrow sinx-\sqrt{3}cosx=\sqrt{2}cos3x+\sqrt{2}sin3x\)

\(\Leftrightarrow\dfrac{1}{2}sinx-\dfrac{\sqrt{3}}{2}cosx=\dfrac{1}{\sqrt{2}}cos3x+\dfrac{1}{\sqrt{2}}sin3x\)

\(\Leftrightarrow sin\left(x-\dfrac{\pi}{3}\right)=sin\left(3x+\dfrac{\pi}{4}\right)\)

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

\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{7\pi}{24}-k\pi\\x=-\dfrac{3}{4}x+\dfrac{13\pi}{48}+\dfrac{k\pi}{2}\end{matrix}\right.\)

Vậy phương trình đã cho có nghiệm \(x=-\dfrac{7\pi}{24}-k\pi;x=-\dfrac{3}{4}x+\dfrac{13\pi}{48}+\dfrac{k\pi}{2}\)

2.

\(sinx-\sqrt{3}cosx=2sin5\text{​​}x\)

\(\Leftrightarrow\dfrac{1}{2}sinx-\dfrac{\sqrt{3}}{2}cosx=sin5x\)

\(\Leftrightarrow sin\left(x-\dfrac{\pi}{3}\right)=sin5x\)

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

\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{12}-\dfrac{k\pi}{2}\\x=\dfrac{2\pi}{9}+\dfrac{k\pi}{3}\end{matrix}\right.\)

Vậy phương trình đã cho có nghiệm \(x=-\dfrac{\pi}{12}-\dfrac{k\pi}{2};x=\dfrac{2\pi}{9}+\dfrac{k\pi}{3}\)

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

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

e ms lớp 8

ai k minh giai cho

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

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

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

a/

\(\Leftrightarrow\left[{}\begin{matrix}cos2x+1=0\\cos2x+2=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}cos2x=-1\\cos2x=-2\left(vn\right)\end{matrix}\right.\)

\(\Rightarrow2x=\pi+k2\pi\)

\(\Rightarrow x=\frac{\pi}{2}+k\pi\)

b/

\(\Leftrightarrow cos5x=sin40^0\)

\(\Leftrightarrow cos5x=cos50^0\)

\(\Leftrightarrow\left[{}\begin{matrix}5x=50^0+k360^0\\5x=-50^0+k360^0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=10^0+k72^0\\x=-10^0+k72^0\end{matrix}\right.\)

c/

\(\Leftrightarrow sin3x=-cosx\)

\(\Leftrightarrow sin3x=sin\left(x-\frac{\pi}{2}\right)\)

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

\(\Rightarrow\left[{}\begin{matrix}x=-\frac{\pi}{4}+k\pi\\x=\frac{3\pi}{8}+\frac{k\pi}{2}\end{matrix}\right.\)

d/

\(\Leftrightarrow2sinx.cosx+\sqrt{3}sinx=0\)

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

\(\Leftrightarrow\left[{}\begin{matrix}sinx=0\\cosx=-\frac{\sqrt{3}}{2}=cos\left(\frac{5\pi}{6}\right)\end{matrix}\right.\)

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

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

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

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

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

\(\Leftrightarrow...\)