Question

Решите уравнение 2sin( п/2 - x) = ctg x

Answer 1

$2\sin\left( \dfrac{\pi}{2} - x\right) = \mathrm{ctg} x$

ОДЗ: $x\neq \pi n,\ n\in\mathbb{Z}$

$2\cos x = \mathrm{ctg} x$

$2\cos x - \mathrm{ctg} x=0$

$2\cos x - \dfrac{\cos x}{\sin x} =0$

$\cos x\left(2 - \dfrac{1}{\sin x}\right)=0$

$\cos x=0\Rightarrow x_1=\dfrac{\pi}{2}+\pi n,\ n\in\mathbb{Z}$

$2 - \dfrac{1}{\sin x}=0\Rightarrow \sin x= \dfrac{1}{2}\Rightarrow x_2=(-1)^k\dfrac{\pi}{6} +\pi k,\ k\in\mathbb{Z}$

Ответ: $\dfrac{\pi}{2}+\pi n;\ (-1)^k\dfrac{\pi}{6} +\pi k,\ n, k\in\mathbb{Z}$