Question

Решить уравнение: -cosx + cos x/2 = 1

Answer 1

$-cosx+cos \frac{x}{2} =1 \\ -2cos^{2} \frac{x}{2} +1+cos \frac{x}{2} =1 \\ -2cos^{2} \frac{x}{2} +cos \frac{x}{2} =0 \\ cos \frac{x}{2} (-2cos \frac{x}{2} +1)=0 \\ 1)cos \frac{x}{2} =0; \frac{x}{2} = \frac{ \pi }{2} + \pi k;x= \pi +2 \pi k; \\ 2)-2cos \frac{x}{2} +1=0 \\ -2cos \frac{x}{2} =-1 \\ cos \frac{x}{2} = \frac{1}{2} \\ \frac{x}{2} =+-arccos \frac{1}{2} +2 \pi n \\ \frac{x}{2} =+- \frac{ \pi }{3} +2 \pi n \\ x=+-\frac{ 2\pi }{3} +4 \pi n \\$

$otvet:\pi +2 \pi k;+- \frac{2 \pi }{3} +4 \pi n;$
k∈Z;n∈Z

Answer 2

-cosx+cosx/2=1
1-2cos²x/2+cosx/2-1=0
-2cos²x/2+cosx/2=0
cosx/2(1-2cosx/2)=0
cosx/2=0⇒x/2=π/2+πn⇒x=π+2πn
cosx/2=1/2⇒x/2=+-π/3+2πn⇒x=+-2π/3+4πn