nierówności trygonometryczne

[olagaw4]

a) sin⁡(1/3 x+π/3)<-√3/2
b) cos⁡3x<1/2
c)〖-sin〗^2 x+sinx+2>0 x∈<0 ,2π>

[eresh]

\(\sin(\frac{\pi}{3}+\frac{x}{3})<-\frac{\sqrt{3}}{2}\\
\frac{\pi}{3}+\frac{x}{3}\in (-\frac{2\pi}{3}+2k\pi,-\frac{\pi}{3}+2k\pi)\\
\frac{x}{3}\in (-\pi+2k\pi, -\frac{2\pi}{3}+2k\pi)\\
x\in (-3\pi+6k\pi, -2\pi+6k\pi)
\)

[eresh]

\(\cos 3x<\frac{1}{2}\\
3x\in (\frac{\pi}{3}+2k\pi, \frac{5\pi}{3}+2k\pi)\\
x\in(\frac{\pi}{9}+\frac{2k\pi}{3},\frac{5\pi}{9}+\frac{2k\pi}{3})\)

[eresh]

c)〖-sin〗^2 x+sinx+2>0 x∈<0 ,2π>

\((-\sin x)^2+\sin x+2>0,x\in [0,2\pi]\\
\sin^2 x+\sin x+2>0,x\in [0,2\pi]\\
t=\sin x\\
t^2+t+2>0\\
t\in \mathbb{R}\\
\sin x\in\mathbb{R}\\
x\in \mathbb{R}\)