Question

|sin(x)|=cos(x)
Решить уравнение

Answer 1

$|\sin x|=\cos x$

Раскроем модуль:

$\left[\begin{array}{l} \left\{\begin{array}{l} \sin x=\cos x \\ \sin x\geq 0 \end{array}\\ \left\{\begin{array}{l} -\sin x=\cos x \\ \sin x<0 \end{array}\end{array}$

$\left[\begin{array}{l} \left\{\begin{array}{l} \mathrm{tg} x=1 \\ \sin x\geq 0 \end{array}\\ \left\{\begin{array}{l} \mathrm{tg} x=-1 \\ \sin x<0 \end{array}\end{array}$

$\left[\begin{array}{l} \left\{\begin{array}{l} x=\frac{\pi }{4}+\pi k \\ \sin x\geq 0 \end{array}\\ \left\{\begin{array}{l} x=-\frac{\pi }{4}+\pi k \\ \sin x<0 \end{array}\end{array}$

$\left[\begin{array}{l} x=\frac{\pi }{4}+2\pi k \\ x=-\frac{\pi }{4}+2\pi k \end{array}$

$x=\pm\dfrac{\pi }{4}+2\pi k, \ k\in\mathbb{Z}$

Ответ: $\pm\dfrac{\pi }{4}+2\pi k, \ k\in\mathbb{Z}$