Trigonometric equations and inequalities

2000001902

Level: 
A
The picture shows the graphical solution of an equation. Which equation is it?
\[ \cos{x} = -\frac{\sqrt{3}}{2}\] \[ x \in [ 0; 2\pi ] \]
\[ \sin{x} = -\frac{\sqrt{3}}{2}\] \[ x \in [ 0; 2\pi ] \]
\[ \sin{x} = -\frac{1}{2}\] \[ x \in [ 0; 2\pi ] \]
\[ \cos{x} = -\frac{1}{2}\] \[ x \in [ 0; 2\pi ] \]

2000001901

Level: 
A
The picture shows the graphical solution of an equation. Which equation is it?
\[ \cos{x} = -\frac{1}{2} \] \[ x \in [ 0;2\pi ]\]
\[ \sin{x} = -\frac{1}{2} \] \[ x \in [ 0;2\pi ]\]
\[ \cos{x} = -\frac{\sqrt{3}}{2} \] \[ x \in [ 0;2\pi ]\]
\[ \sin{x} = -\frac{\sqrt{3}}{2} \] \[ x \in [ 0;2\pi ]\]

1003085910

Level: 
C
The solution set of the inequality \( \mathrm{tg}^3x + \mathrm{tg}^2x - \mathrm{tg}\,x - 1 < 0 \) for \( x\in\left(-\frac{\pi}2;\frac{\pi}2\right) \) is:
\( \left(-\frac{\pi}2;\frac{\pi}4\right) \)
\( \left(\frac{\pi}2;\frac{7\pi}4\right) \)
\( \left(-\frac{\pi}2;\frac{3\pi}4\right) \)
\( \left(-\frac{\pi}2;\frac{\pi}2\right) \)

1003085909

Level: 
C
The solution set of the inequality \( |\mathrm{tg}\,x| < 1 \) for \( x\in\left(-\frac{\pi}2;\frac{\pi}2\right) \) is:
\( \left( -\frac{\pi}4;\frac{\pi}4 \right) \)
\( \left( -\frac{\pi}2;\frac{\pi}2 \right) \)
\( \left( 0;\frac{\pi}4 \right) \)
\( \left( -\frac{\pi}2;-\frac{\pi}4 \right) \)

1003085908

Level: 
C
The solution set of the inequality \( \mathrm{cotg}\left(3x -\frac{\pi}4 \right) \geq -1 \) for \( x\in\left[0;\frac{\pi}2\right] \) is:
\( \{0\}\cup\left( \frac{\pi}{12};\frac{\pi}3 \right]\cup\left(\frac{5\pi}{12};\frac{\pi}2\right] \)
\( \left[ 0;\frac{\pi}3\right] \)
\( \left( \frac{\pi}{12};\frac{\pi}3\right] \)
\( \left( 0;\frac{\pi}2\right] \)

1003085907

Level: 
C
The solution set of the inequality \( |\sin x| \geq 0.5 \) for \( x\in[0;2\pi] \) is:
\( \left[\frac{\pi}6;\frac{5\pi}6\right]\cup\left[\frac{7\pi}6;\frac{11\pi}6\right] \)
\( \left[0;\frac{5\pi}6\right]\cup\left[\frac{7\pi}6;\frac{11\pi}6\right] \)
\( \left[\frac{\pi}6;\frac{11\pi}6\right] \)
\( \left[0;\frac{\pi}6\right]\cup\left[\frac{5\pi}6;\frac{11\pi}6\right] \)

1003085906

Level: 
C
The solution set of the inequality \( \cos 3x > \frac{{\sqrt3}}2 \) for \( x\in(0^{\circ};180^{\circ}) \) is:
\( \left(0^{\circ};10^{\circ} \right)\cup\left(110^{\circ};130^{\circ}\right) \)
\( \left(330^{\circ};360^{\circ}\right) \)
\(\left(110^{\circ};120^{\circ}\right) \)
\( \left(0^{\circ};30^{\circ} \right) \)

1003085905

Level: 
C
The solution set of the inequality \( |\mathrm{cotg} x |\leq \sqrt3 \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}6+k\pi;\frac{5\pi}6+k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}6+k\pi;\frac{\pi}2+k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}2+k\pi;\frac{5\pi}6+k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[ k\pi;(k+1)\pi\right] \)

1003085903

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