Question

Giải các phương trình:

\(a,sinx-cosx=1\)

\(b,cos3x.cos2x=cos5x\)

Answer 1

a, \(sinx-cosx=1\)

\(\Leftrightarrow\sqrt{2}sin\left(x-\dfrac{\pi}{4}\right)=1\)

\(\Leftrightarrow sin\left(x-\dfrac{\pi}{4}\right)=\dfrac{1}{\sqrt{2}}\)

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

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

Answer 2

b, \(cos3x.cos2x=cos5x\)

\(\Leftrightarrow\dfrac{1}{2}cos5x+\dfrac{1}{2}cosx=cos5x\)

\(\Leftrightarrow cosx=cos5x\)

\(\Leftrightarrow5x=\pm x+k2\pi\)

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