Question

ПОМОГИТЕ С ТРИГОНОМЕТРИЕЙ, пожалуйста. 2 и 3 номера.

Answer 1

2а)
$\sin(x - \frac{\pi}{3} ) \geqslant \frac{ \sqrt{3} }{2} \\ \frac{ \pi}{3} + 2\pi \times n \leqslant x - \frac{\pi}{3} \leqslant \frac{2\pi}{3} + 2\pi \times n \\ \frac{ \pi}{3} +\frac{ \pi}{3} + 2\pi \times n \leqslant x - \frac{\pi}{3} + \frac{ \pi}{3}\leqslant \frac{2\pi}{3} +\frac{ \pi}{3} + 2\pi \times n \\ \frac{ 2\pi}{3} + 2\pi \times n \leqslant x\leqslant \pi+ 2\pi \times n$
nєZ.
xє[2π/3 + 2πn; π+2πn], nєZ.

2b)
$\cos(2x + \frac{\pi}{4} ) \leqslant - \frac{ \sqrt{2} }{2} \\ \frac{3\pi}{4} + 2\pi \times n \leqslant 2x + \frac{\pi}{4} \leqslant \frac{5\pi}{4} + 2\pi \times n \\ \frac{3\pi}{4} - \frac{\pi}{4} + 2\pi \times n \leqslant 2x + \frac{\pi}{4} - \frac{\pi}{4} \leqslant \frac{5\pi}{4} - \frac{\pi}{4} + 2\pi \times n \\ \frac{\pi}{2} + 2\pi \times n \leqslant 2x \leqslant \pi+ 2\pi \times n \: \: \: \: ( \div 2) \\ \frac{\pi}{4} + \pi \times n \leqslant x \leqslant \frac{\pi}{2} + \pi \times n$
nєZ.
xє[π/4 +πn; π/2+πn], nєZ.

3.
$2 \sin^{2}(x) - \cos(x) > 2 \\ \sin^{2}(x) = 1 - \cos^{2}(x) \\ 2(1 - \cos^{2}(x)) - \cos(x) > 2 \\ 2 - 2\cos^{2}(x) - \cos(x) > 2 \\ 2 - 2\cos^{2}(x) - \cos(x) - 2 > 0 \\ - 2\cos^{2}(x) - \cos(x) > 0 \: \: \: (\times - 1) \\ 2\cos^{2}(x) + \cos(x) < 0 \\ \cos(x) (2 \cos(x) + 1) < 0$
найдем нули функции
соs(x)=0 при х1=π/2 +2πn, nєZ,
x2=3π/2 +2πn, nєZ.
2cos(x)+1=0
cos(x)=-1/2
x1=2π/3 +2πn, nєZ,
x2=4π/3+2πn, nєZ.

___o_____o_____o_____o____
..+..π/2...-...2π/3..+..4π/3...-...3π/2..+.

xє(π/2+2πn;2π/3+2πn)U(4π/3+2πn;3π/2+2πn), nєZ.