Trigonometric equations and inequalities

1003085804

Level: 
B
The solution set of the inequality \( \sin x > \cos x \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left(\frac{\pi}4+2k\pi;\ \frac{5\pi}4+2k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(\frac{\pi}4+k\pi;\ \frac{5\pi}4+k\pi \right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(\frac{\pi}3+2k\pi;\ \frac{5\pi}3+2k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(\frac{\pi}3+k\pi;\ \frac{5\pi}3+k\pi \right) \)

1003085803

Level: 
B
The solution set of the inequality \( \mathrm{cotg}\,x \leq \sqrt3 \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}6+k\pi;\ (k+1)\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}6+2k\pi;\ \frac{\pi}2+2k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}3+k\pi;\ \frac{\pi}2+k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}3+2k\pi;\ \frac{\pi}2+2k\pi\right] \)

1003085802

Level: 
B
The solution set of the inequality \( \mathrm{tg}\, x \leq \frac{\sqrt3}3 \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left(-\frac{\pi}2+k\pi;\ \frac{\pi}6+k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(\frac{\pi}6+k\pi;\ \frac{\pi}3+k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(-\frac{\pi}2+2k\pi;\ \frac{\pi}6+2k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(\frac{\pi}6+2k\pi;\ \frac{\pi}3+2k\pi\right) \)

1003085801

Level: 
B
The solution set of the inequality \( \cos x > 0.5 \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left(-\frac{\pi}3+2k\pi;\ \frac{\pi}3+2k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(-\frac{\pi}3+k\pi;\ \frac{\pi}3+k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(-\frac{\pi}3+2k\pi;\ k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(-\frac{\pi}3+2k\pi;\ 2k\pi\right) \)

1003085705

Level: 
A
Solving the equation \( 2\sin\!\left(x + \frac{\pi}4 \right) = \sqrt3 \) for \( x \), where \( x\in (0; \pi) \), you get:
\( x\in\left\{ \frac{\pi}{12};\frac{5\pi}{12} \right\} \)
\( x\in\left\{ \frac{\pi}{12} \right\} \)
\( x\in\left\{ \frac{3\pi}{12};\frac{5\pi}{12} \right\} \)
\( x\in\left\{ \frac{13\pi}{12};\frac{5\pi}{12} \right\} \)