Trigonometric equations and inequalities

2010014905

Level: 
B
Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation. \[ \mathop{\mathrm{tg}}^2\nolimits x - 2\mathop{\mathrm{tg}}\nolimits x -3=0 \]
substitution \( \mathop{\mathrm{tg}}\nolimits x =y\)
\(\mathop{\mathrm{tg}}\nolimits x (\mathop{\mathrm{tg}}\nolimits x -2)=3\)
\(\frac{\sin^2 x}{\cos^2 x}-2\frac{\sin x}{\cos x}-3=0\)
\(\mathop{\mathrm{tg}}\nolimits x-2=3-\mathop{\mathrm{tg}}\nolimits x \)

2010014904

Level: 
B
Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation. \[ 3 \cos^2 x =2\sin x \cos x \]
\(\cos x (3\cos x-2\sin x)=0\)
\(3\cos x=2\sin x\)
\(3(1-\sin^2 x)=2\sin x \cos x\)
\(\frac{3\cos^2 x}{2\sin x \cos x}=1\)

2010014903

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

2010014902

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

2010014901

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

2010012002

Level: 
A
Solve \( \cos^2x = \sqrt2 \cos x \) for \( x \), where \( x\in\mathbb{R} \).
\( x\in\bigcup\limits_{k\in\mathbb{Z}} \left\{ \frac{\pi}2+k\pi \right\} \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}} \left\{ \frac{\pi}4+k\pi ;\frac{\pi}2+k\pi\right\} \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}} \left\{ \frac{\pi}4+k\pi \right\} \)
\( x \in \emptyset \)

2010012001

Level: 
A
Find all \( x \), \( x\in\mathbb{R} \), such that \( \mathrm{tg}^2x = \mathrm{tg}\,x \).
\( x\in\bigcup\limits_{k\in\mathbb{Z}} \left\{k\pi;\frac{\pi}4+k\pi \right\} \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}} \left\{k\pi\right\} \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}4+k\pi \right\} \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}} \left\{\frac{\pi}2+k\pi;\frac{\pi}4+k\pi \right\} \)