Question

Как решить это уравнение

Answer 1

24) 2sinx + cos^2(x/2) + 2 + cosx = 0
cos^2(x/2) = (1 + cosx)/2
2sinx + (1 + cosx)/2 + 2 + cosx = 0 - умножим на 2
4sinx + 1 + cosx + 4 + 2cosx = 0
4sinx + 3cosx + 5 = 0
sinx = sin(2*x/2) = 2sin(x/2)*cos(x/2)
cos(2*x/2) = cos^2(x/2) - sin^2(x/2)
8sin(x/2)*cos(x/2) + 3cos^2(x/2) - 3sin^2(x/2) + 5sin^2(x/2) + 5cos^2(x/2) = 0
4sin(x/2)*cos(x/2) + 4cos^2(x/2) + sin^2(x/2) = 0
sin^2(x/2) + 2*2sin(x/2)*cos(x/2) + 4cos^2(x/2) = 0
(sin(x/2) + 2cos(x/2))^2 = 0
sin(x/2) + 2cos(x/2) = 0
sin(x/2) = -2cos(x/2)
tg(x/2) = -2
(x/2) = +-arctg(-2) + pi*k
x = +-2arctg(2) + 2pi*k

25) cos(2x) = -1 - sin(2x)
cos(2x) = cos^2(x) - sin^2(x)
sin(2x) = 2sinx*cosx
1 = cos^2(x) + sin^2(x)
cos^2(x) - sin^2(x) + cos^2(x) + sin^2(x) + 2sinx*cosx = 0
2cos^2(x) + 2sinx*cosx = 0
cosx*(1 + sinx) = 0
a) cosx = 0, x = pi/2 + pi*k
b) sinx = -1, x = -pi/2 + 2pi*k