삼각방정식 근의 합

반각공식과 배각공식을 이용하면 주어진 방정식은 $\displaystyle 6 \times \frac{1 - \cos 2 x}{2} + \frac{\sqrt{3}}{2} \times \sin 2 x + 7 \times \frac{1 + \cos 2 x}{2} = 6$ $\displaystyle \sqrt{3} \sin 2 x + \cos 2 x = - 1$$\leftarrow$합성하면 $\displaystyle \sin \left( 2 x + \frac{\pi}{6} \right) = - \frac{1}{2}$ $( - \pi \leq x \leq \pi )$에서 $\displaystyle \theta = 2 x + \frac{\pi}{6}$라 두면 $\displaystyle - 2 \pi + \frac{\pi}{6} \leq \theta \leq 2 \pi + \frac{\pi}{6}$이므로 (그래프를 이용하면) $\displaystyle \theta = - \pi + \frac{\pi}{6} , - \frac{\pi}{6} , \pi + \frac{\pi}{6} , 2 \pi - \frac{\pi}{6}$ $\displaystyle \therefore x = - \frac{\pi}{2} , - \frac{\pi}{6} , \frac{\pi}{2} , \frac{5 \pi}{6}$ $\displaystyle \therefore \text{모든} \text{합은} \frac{2 \pi}{3}$