Trigonometric equations and inequalities

2010009804

Level: 
C
The solution set of the equation \( \mathrm{tg}\, x - \mathrm{cotg}\,x = 0 \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}4+k\pi;\frac{3\pi}4+k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{k\pi;\frac{\pi}4+k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}4+k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{3\pi}4+k\pi\right\} \)

2000006604

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

2000006603

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

2000006602

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

2000006601

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

2000006404

Level: 
A
An equation is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding equation.
\[ \mathrm{cotg}\,{x} = 1\] \[ x \in ( -\pi ;2\pi)\]
\[ \mathrm{cotg}\,{x} = 1\] \[ x \in (0 ;2\pi )\]
\[ \mathrm{cotg}\,{x} = \frac{\sqrt{3}}{2} \] \[ x \in ( -\pi ;2\pi)\]
\[ \mathrm{cotg}\,{x} = \frac{\sqrt{3}}{2} \] \[ x \in ( 0 ;2\pi)\]

2000006403

Level: 
A
An equation is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding equation.
\[ \mathrm{cotg}\,{x} = -\frac{\sqrt{3}}{3} \] \[ x \in ( -\pi ;2\pi)\]
\[ \mathrm{cotg}\,{x} = -\frac{1}{2} \] \[ x \in (-\pi ;2\pi )\]
\[ \mathrm{tg}\,{x} = -\frac{\sqrt{3}}{2} \] \[ x \in ( -\pi ;2\pi)\]
\[ \mathrm{tg}\,{x} = -\frac{1}{2} \] \[ x \in ( -\pi ;2\pi)\]