Trigonometric equations and inequalities

1003085903

Level: 
B
The solution set of the inequality \( \sin^2x\leq1 \) for \( x\in\mathbb{R} \) is:
\( \mathbb{R} \)
\( \emptyset \)
\( \mathbb{R}\setminus\left\{\frac{\pi}2+2k\pi\colon k\in\mathbb{Z}\right\} \)
\( \mathbb{R}\setminus\left\{\frac{3\pi}2+2k\pi\colon k\in\mathbb{Z}\right\} \)

1003086008

Level: 
B
The solution set of the equation \( \mathrm{tg}\,x\cdot\mathrm{cotg}\,x = 1 \) for \( x\in\mathbb{R} \) is:
\( \mathbb{R}\setminus\left\{\frac{k\pi}2\colon k\in\mathbb{Z}\right\} \)
\( \mathbb{R} \)
\( \mathbb{R}\setminus\left\{k\pi\colon k\in\mathbb{Z}\right\} \)
\( \mathbb{R}\setminus\left\{2k\pi\colon k\in\mathbb{Z}\right\} \)

1003086106

Level: 
B
The solution set of the equation \( \sin 2x = \cos 3x \cdot \sin 2x \) for \( x\in\left[0^{\circ};180^{\circ}\right] \) is:
\( \left\{0^{\circ};90^{\circ};120^{\circ};180^{\circ}\right\} \)
\( \left\{90^{\circ};120^{\circ};180^{\circ}\right\} \)
\( \left\{90^{\circ};180^{\circ}\right\} \)
\( \left\{0^{\circ};90^{\circ}\right\} \)

1003086109

Level: 
B
The solution set of the equation \( \mathrm{tg}\,x + \mathrm{cotg}\,x = 2 \) for \( x\in[-2\pi;2\pi] \) is:
\( \left\{-\frac{7\pi}4;-\frac{3\pi}4;\frac{\pi}4;\frac{5\pi}4 \right\} \)
\( \left\{-\frac{7\pi}4;\frac{\pi}4;\right\} \)
\( \left\{-\frac{5\pi}4;-\frac{\pi}4;\frac{3\pi}4;\frac{7\pi}4 \right\} \)
\( \left\{-\frac{3\pi}4;\frac{\pi}4\right\} \)

2000006301

Level: 
B
An inequality is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding inequality.
\[ \sin{x} \geq \frac{\sqrt{3}}{2} \] \[ x \in [ 0;2\pi ]\]
\[ \sin{x} \geq \frac{\sqrt{2}}{2} \] \[ x \in [ 0;2\pi ]\]
\[ \sin{x} > \frac{\sqrt{3}}{2} \] \[ x \in [ 0;2\pi ]\]
\[ \sin{x} > \frac{\sqrt{2}}{2} \] \[ x \in [ 0;2\pi ]\]