Question

Solve the following equation:
4sinx cosx + 2 sin x + 2 cosx + 1 = 0 

Answer 1

\[4 \sin x \cos x + 2 \sin x + 2 \cos x + 1 = 0\]
\[ \Rightarrow 2 \sin x\left( 2 \cos x + 1 \right) + 1\left( 2 \cos x + 1 \right) = 0\]
\[ \Rightarrow \left( 2 \sin x + 1 \right)\left( 2 \cos x + 1 \right) = 0\]
\[ \Rightarrow 2 \sin x + 1 = 0\text{ or }2 \cos x + 1 = 0\]
\[ \Rightarrow \sin x = - \frac{1}{2} \text{ or }\cos x = - \frac{1}{2}\]
\[ \Rightarrow \sin x = \sin\frac{7\pi}{6}\text{ or }\cos x = \frac{2\pi}{3}\]
\[ \Rightarrow x = n\pi + \left( - 1 \right)^n \frac{7\pi}{6}\text{ or }x = 2n\pi \pm \frac{2\pi}{3}, n \in \mathbb{Z}\]