Question

Помогите решить простейшие тригонометрические уравнения:
1) cos x/5= -7Пи/22;
2) tg(x +2)=0;
3) sinПи√x =-1;
4) 4cos(Пи/3 - 1/2x)=0

Answer 1

$1)\quad cos \frac{x}{5}=-\frac{7\pi}{22}\\\\\frac{x}{5}=\pm arccos(-\frac{7\pi}{22})+2\pi n=\pm (\pi -arccos\frac{7\pi}{22})+2\pi n,\\\\x=\pm (5\pi -5arccos\frac{7\pi}{22})+10\pi n,\; n\in Z\\\\2)\quad tg(x+2)=0\\\\x+2=\pi n,\\\\x=-2+\pi n,\; n\in Z\\\\3)\quad sin(\pi \sqrt{x})=-1\\\\\pi \sqrt{x}=-\frac{\pi }{2}+2\pi n,\\\\\sqrt{x}=-\frac{1}{2}+2n,\\\\x=(2n-\frac{1}{2})^2\; ,\; n\in Z \\\\4)\quad 4cos(\frac{\pi}{3}-\frac{1}{2}x)=0\\\\cos( \frac{\pi }{3} - \frac{1}{2}x)=0\\\\ \frac{\pi }{3} - \frac{1}{2}x= \frac{\pi}{2}+\pi n,$

$\frac{1}{2}x=\frac{\pi}{3}-\frac{\pi}{2}-\pi n=-\frac{\pi}{6}-\pi n,\\\\x=-\frac{\pi}{3}-2\pi n,\; n\in Z$