Question

\(\cos\left(x+\frac{\pi}{5}\right)+\sqrt{3}sin\left(x+\frac{\pi}{5}\right)=0\left(x\in\text{[}-2020\pi;\frac{\pi}{2}\text{]}\right)\)

Answer 1

\(\frac{1}{2}.cos\left(x+\frac{\pi}{5}\right)+\frac{\sqrt{3}}{2}.sin\left(x+\frac{\pi}{5}\right)=0\)

\(\Leftrightarrow cos\left(\frac{\pi}{3}\right).cos\left(x+\frac{\pi}{5}\right)+sin\left(\frac{\pi}{3}\right).sin\left(x+\frac{\pi}{5}\right)=0\)

\(\Leftrightarrow cos\left(x+\frac{\pi}{5}-\frac{\pi}{3}\right)=0\)

\(\Leftrightarrow cos\left(x-\frac{2\pi}{15}\right)=0\)

\(\Leftrightarrow x-\frac{2\pi}{15}=\frac{\pi}{2}+k\pi\)

\(\Leftrightarrow x=\frac{19\pi}{30}+k\pi\) \(k\in\left[-2020;-1\right]\)