Trigonometric equations and inequalities

2000006302

Level: 
B
An inequality is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding inequality.
\[ \sin{x} < \frac{1}{2} \] \[ x \in [ 0;2\pi ]\]
\[ \sin{x} \leq \frac{\sqrt{2}}{2} \] \[ x \in [ 0;2\pi ]\]
\[ \sin{x} < \frac{\sqrt{2}}{2} \] \[ x \in [ 0;2\pi ]\]
\[ \sin{x} \leq \frac{1}{2} \] \[ x \in [ 0;2\pi ]\]

2000006303

Level: 
B
An inequality is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding inequality.
\[ \cos{x} < \frac{1}{2} \] \[ x \in [ 0;2\pi ]\]
\[ \cos{x} \leq \frac{\sqrt{2}}{2} \] \[ x \in [ 0;2\pi ]\]
\[ \cos{x} < \frac{\sqrt{2}}{2} \] \[ x \in [ 0;2\pi ]\]
\[ \cos{x} \leq \frac{1}{2} \] \[ x \in [ 0;2\pi ]\]

2000006304

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

2000006601

Level: 
B
An inequality is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding inequality.
\[ \mathrm{tg}\,{x} \geq \frac{\sqrt{3}}{3} \] \[ x \in [ 0 ;\pi ] \setminus \left\{ \frac{\pi}{2} \right\}\]
\[ \mathrm{tg}\,{x} \geq \frac{\sqrt{3}}{2} \] \[ x \in [ 0 ;\pi ] \setminus \left\{ \frac{\pi}{2} \right\}\]
\[ \mathrm{cotg}\,{x} \geq \frac{\sqrt{3}}{2} \] \[ x \in [ 0 ;\pi ] \setminus \left\{ \frac{\pi}{2} \right\}\]
\[ \mathrm{cotg}\,{x} \geq \frac{\sqrt{3}}{3} \] \[ x \in [ 0 ;\pi ] \setminus \left\{ \frac{\pi}{2} \right\}\]

2000006602

Level: 
B
An inequality is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding inequality.
\[ \mathrm{tg}\,{x} \leq -\sqrt{3} \] \[ x \in [ -\pi ;\pi ] \setminus \left\{ -\frac{\pi}{2};\frac{\pi}{2} \right\}\]
\[ \mathrm{tg}\,{x} \geq -\sqrt{3} \] \[ x \in [ -\pi ;\pi ] \setminus \left\{ -\frac{\pi}{2};\frac{\pi}{2} \right\}\]
\[ \mathrm{cotg}\,{x} \leq -\sqrt{3} \] \[ x \in [ -\pi ;\pi ] \setminus \left\{ -\frac{\pi}{2};\frac{\pi}{2} \right\}\]
\[ \mathrm{cotg}\,{x} \geq -\sqrt{3} \] \[ x \in [ -\pi ;\pi ] \setminus \left\{ -\frac{\pi}{2};\frac{\pi}{2} \right\}\]

2000006603

Level: 
B
An inequality is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding inequality.
\[ \mathrm{cotg}\,{x} \leq 1 \] \[ x \in (-\pi ;\pi ) \setminus \left\{ 0\right\}\]
\[ \mathrm{cotg}\,{x} \geq 1 \] \[ x \in (-\pi ;\pi ) \setminus \left\{ 0\right\}\]
\[ \mathrm{tg}\,{x} \leq 1\] \[ x \in (-\pi ;\pi ) \setminus \left\{ 0\right\}\]
\[ \mathrm{tg}\,{x} \geq 1\] \[ x \in (-\pi ;\pi ) \setminus \left\{ 0\right\}\]

2000006604

Level: 
B
An inequality is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding inequality.
\[ \mathrm{cotg}\,{x} \geq -\frac{\sqrt{3}}{3}\] \[ x \in (-\pi ;\pi ) \setminus \left\{ 0\right\}\]
\[ \mathrm{cotg}\,{x} \geq \frac{1}{2} \] \[ x \in (-\pi ;\pi ) \setminus \left\{ 0\right\}\]
\[ \mathrm{cotg}\,{x} \geq \frac{\sqrt{3}}{2}\] \[ x \in (-\pi ;\pi ) \setminus \left\{ 0\right\}\]
\[ \mathrm{cotg}\,{x} \leq \frac{\sqrt{3}}{3}\] \[ x \in (-\pi ;\pi ) \setminus \left\{ 0\right\}\]

2010014901

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

2010014902

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

2010014903

Level: 
B
The solution set of the inequality \( \cos\,x < -\frac{\sqrt3}{2} \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left(\frac{5\pi}6+2k\pi;\frac{7\pi}6+2k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(\frac{5\pi}6+k\pi;\frac{7\pi}6+k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(\frac{5\pi}6+2k\pi;\frac{11\pi}6+2k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(-\frac{5\pi}6+2k\pi;\frac{5\pi}6+2k\pi\right) \)