Question

Решить уравнение 1)sin(x/3 - пи/4)=-1\2 2)cos 4x = √3\2

Answer 1

1)

$\displaystyle sin(\frac{x}{3}-\frac{\pi}{4})= -\frac{1}{2}\\\\\ \left[\begin{array}{ccc}\frac{x}{3}-\frac {\pi}{4}=-\frac{\pi}{6}+2\pi n; n\in Z \\ \frac{x}{3}-\frac{\pi }{4} =-\frac{5\pi }{6}+2\pi n; n\in Z\\ \end{array}\\\\$

$\left[\begin{array}{ccc}\frac{x}{3}=+\frac {\pi}{4}-\frac{\pi}{6}+2\pi n; n\in Z \\ \frac{x}{3}=+\frac{\pi }{4}-\frac{5\pi }{6}+2\pi n; n\in Z\\\end{array}\\\\$

$\left[\begin{array}{ccc}\frac{x}{3}=\frac{\pi}{12}+2\pi n; n\in Z \\\frac{x}{3}=-\frac{7\pi }{12}+2\pi n; n\in Z\\\end{array}\\\\$

$\left[\begin{array}{ccc}x=\frac{\pi}{4}+6\pi n; n\in Z \\x =-\frac{7\pi }{4}+6\pi n; n\in Z\\ \end{array}\\\\$

2)

$\displaystyle cos(4x)=\frac{\sqrt{3}}{2}\\\\4x= \pm \frac{\pi }{6} +2\pi n; n\in Z\\\\ x=\pm\frac{\pi}{24}+\frac{\pi n}{2} ; n\in Z$