Question

Решить уравнение 2cos(x/5-π/3)=√3

Answer 1

$2\cos\left(\dfrac{x}{5} -\dfrac{\pi }{3} \right)=\sqrt{3}$

$\cos\left(\dfrac{x}{5} -\dfrac{\pi }{3} \right)=\dfrac{\sqrt{3} }{2}$

$\dfrac{x}{5} -\dfrac{\pi }{3} =\pm\dfrac{\pi }{6} +2\pi n$

$\dfrac{x}{5} =\dfrac{\pi }{3}\pm\dfrac{\pi }{6} +2\pi n$

$\boxed{x =\dfrac{5\pi }{3}\pm\dfrac{5\pi }{6} +10\pi n.\ n\in\mathbb{Z}}$

Или, записывая в виде отдельных серий корней:

$\left[\begin{array}{l} x_1 =\dfrac{5\pi }{3}+\dfrac{5\pi }{6} +10\pi n \\ x_2 =\dfrac{5\pi }{3}-\dfrac{5\pi }{6} +10\pi n \end{array}\right.$

$\left[\begin{array}{l} x_1 =\dfrac{10\pi }{6}+\dfrac{5\pi }{6} +10\pi n \\ x_2 =\dfrac{10\pi }{6}-\dfrac{5\pi }{6} +10\pi n \end{array}\right.$

$\left[\begin{array}{l} x_1 =\dfrac{15\pi }{6} +10\pi n \\ x_2 =\dfrac{5\pi }{6} +10\pi n \end{array}\right.$

$\left[\begin{array}{l} x_1 =\dfrac{5\pi }{2} +10\pi n \\ x_2 =\dfrac{5\pi }{6} +10\pi n \end{array}\right.,\ n\in\mathbb{Z}$