Trigonometric equations and inequalities

1003085702

Level: 
A
Solve \( 2\sin^2x = \sqrt2 \sin x \) for \( x \), where \( x\in\mathbb{R} \).
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left[ \{k\pi\}\cup\left\{ \frac{\pi}4+2k\pi \right\}\cup\left\{ \frac{3\pi}4+2k\pi\right\} \right] \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left[ \left\{ \frac{\pi}4+2k\pi \right\}\cup\left\{ \frac{3\pi}4+2k\pi\right\} \right] \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left[ \left\{ \frac{\pi}4+k\pi \right\}\cup\left\{ \frac{3\pi}4+k\pi\right\} \right] \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left[ \{2k\pi\}\cup\left\{ \frac{\pi}4+k\pi \right\}\cup\left\{ \frac{3\pi}4+k\pi\right\} \right] \)

1003085701

Level: 
A
Find all \( x \), \( x\in\mathbb{R} \), such that \( \mathrm{cotg}^2x = - \mathrm{cotg}\,x \).
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left[ \left\{\frac{3\pi}4+k\pi \right\}\cup\left\{\frac{\pi}2+k\pi \right\} \right] \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{3\pi}4+2k\pi \right\} \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{3\pi}4+k\pi \right\} \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}2+k\pi \right\} \)

1003085610

Level: 
A
The solution set of the equation \( \mathrm{cotg}(2x - \pi) = 1 \) for \( x\in\mathbb{R} \) is:
\( \left\{\frac{5\pi}8 + \frac{k\pi}2\colon k\in\mathbb{Z}\right\} \)
\( \left\{\frac{5\pi}4 + k\pi\colon k\in\mathbb{Z}\right\} \)
\( \left\{\frac{5\pi}8 + k\pi\colon k\in\mathbb{Z}\right\} \)
\( \left\{\frac{5\pi}4 +\frac{k\pi}2\colon k\in\mathbb{Z}\right\} \)

1003085609

Level: 
A
Find the set of all \( x\in\mathbb{R} \) satisfying the equation \( \mathrm{tg}\,(3x - \pi) = 1 \).
\( \left\{\frac{5\pi}{12} + \frac{k\pi}3\colon k\in\mathbb{Z}\right\} \)
\( \left\{\frac{5\pi}{12} + k\pi\colon k\in\mathbb{Z}\right\} \)
\( \left\{\frac{5\pi}4 + k\pi\colon k\in\mathbb{Z}\right\} \)
\( \left\{\frac{5\pi}4 + \frac{k\pi}3\colon k\in\mathbb{Z}\right\} \)

1003085608

Level: 
A
Find the set of all \( x\in\mathbb{R} \) satisfying the equation \( \cos\!\left(2x - \pi\right) = -1\).
\( \left\{ k\pi\colon k\in\mathbb{Z}\right\} \)
\( \left\{\frac{\pi}2 + 2k\pi\colon k\in\mathbb{Z}\right\} \)
\( \left\{2k\pi\colon k\in\mathbb{Z}\right\} \)
\( \left\{ \frac{3\pi}2+ 2k\pi\colon k\in\mathbb{Z}\right\} \)

1003085607

Level: 
A
Solve the equation \( \sin\!\left(4x - \pi\right) = 0 \) for \( x\in\mathbb{R} \).
\( x= \frac{k\pi}4\text{, }k\in\mathbb{Z} \)
\( x=\frac{\pi}4 + \frac{k\pi}2\text{, }k\in\mathbb{Z} \)
\( x= \frac{k\pi}2\text{, }k\in\mathbb{Z} \)
\( x= \frac{\pi}2 + \frac{k\pi}2\text{, }k\in\mathbb{Z} \)

1003085606

Level: 
A
Find the set of all \( x\in\mathbb{R} \) for which \( \sin\!\left(\frac{\pi}6 + x\right) = -0.5 \).
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{5\pi}3+2k\pi;\ \pi+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}2 + 2k\pi;\ \frac{5\pi}3+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{5\pi}3+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{7\pi}6+2k\pi;\ \frac{11\pi}6+2k\pi\right\} \)

1003085605

Level: 
A
Find the set of all \( x\in\mathbb{R} \) for which \( \cos\!\left(\frac{\pi}4 - x\right) = 1 \).
\( \left\{\frac{\pi}4 + 2k\pi\colon k\in\mathbb{Z} \right\} \)
\( \left\{\frac{\pi}2 + 2k\pi\colon k\in\mathbb{Z} \right\} \)
\( \left\{\frac{\pi}4 + k\pi\colon k\in\mathbb{Z} \right\} \)
\( \left\{\frac{\pi}2 + k\pi\colon k\in\mathbb{Z} \right\} \)