Question

cos(4x+pi/3)+sin(x+pi/5)=0

Answer 1

\(\Leftrightarrow cos\left(4x+\dfrac{\pi}{3}\right)=-sin\left(x+\dfrac{\pi}{5}\right)\)

\(\Leftrightarrow cos\left(4x+\dfrac{\pi}{3}\right)=cos\left(x+\dfrac{7\pi}{10}\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}4x+\dfrac{\pi}{3}=x+\dfrac{7\pi}{10}+k2\pi\\4x+\dfrac{\pi}{3}=-x-\dfrac{7\pi}{10}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}3x=\dfrac{11\pi}{30}+k2\pi\\5x=-\dfrac{31\pi}{30}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{11\pi}{90}+\dfrac{k2\pi}{3}\\x=-\dfrac{31\pi}{150}+\dfrac{k2\pi}{5}\end{matrix}\right.\) (\(k\in Z\))