Trigonometric equations and inequalities

1003085903

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

1003085902

Level: 
C
The solution set of the inequality \( \sin^2x \geq 0.75 \) for \( x\in[0; 2\pi] \) is:
\( \left[ \frac{\pi}3;\frac{2\pi}3\right]\cup\left[ \frac{4\pi}3;\frac{5\pi}3\right] \)
\( \left[ \frac{\pi}3;\frac{5\pi}3\right] \)
\( \left[ 0;\frac{2\pi}3\right] \)
\( \left[ 0;\frac{5\pi}3\right] \)

1003085901

Level: 
C
The solution set of the inequality \( \cos^2x\leq0.25 \) for \( x\in[0;2\pi] \) is:
\( \left[ \frac{\pi}3;\frac{2\pi}3\right]\cup\left[ \frac{4\pi}3;\frac{5\pi}3\right] \)
\( \left[ \frac{\pi}3;\frac{5\pi}3\right] \)
\( \left[ 0;\frac{2\pi}3\right] \)
\( \left[ 0;\frac{5\pi}3\right] \)

1003086009

Level: 
C
The solution set of the equation \( \sin x + \cos x = 0 \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{3\pi}4+2k\pi;\frac{7\pi}4+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{3\pi}4+2k\pi\right\} \)
\( \mathbb{R} \)
\( \emptyset \)

1003086008

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

1003086007

Level: 
C
The solution set of the equation \( \sin^4x = 1 - \cos^2x \) for \( x\in\left[0^{\circ};360^{\circ}\right] \) is:
\( \left\{0^{\circ};90^{\circ};180^{\circ};270^{\circ};360^{\circ} \right\} \)
\( \left\{0^{\circ};180^{\circ};360^{\circ} \right\} \)
\( \left\{0^{\circ};90^{\circ};270^{\circ};360^{\circ} \right\} \)
\( \left\{90^{\circ};270^{\circ}\right\} \)