A

1003085705

Část: 
A
Řešením rovnice \( 2\sin\!\left(x + \frac{\pi}4 \right) = \sqrt3 \), kde \( x\in (0; \pi) \), dostaneme:
\( x\in\left\{ \frac{\pi}{12};\frac{5\pi}{12} \right\} \)
\( x\in\left\{ \frac{\pi}{12} \right\} \)
\( x\in\left\{ \frac{3\pi}{12};\frac{5\pi}{12} \right\} \)
\( x\in\left\{ \frac{13\pi}{12};\frac{5\pi}{12} \right\} \)

1003085704

Část: 
A
Řešením rovnice \( \cos\!\left(2x - \frac{\pi}3 \right) = - 0{,}5 \), kde \( 0 < x < 2\pi \), je:
\( \left\{ \frac{\pi}2; \frac{3\pi}2; \frac{5\pi}6; \frac{11\pi}6 \right\} \)
\( \left\{ \frac{\pi}2; \frac{3\pi}2 \right\} \)
\( \left\{ \frac{5\pi}6; \frac{11\pi}6 \right\} \)
\( \left\{ \frac{3\pi}2; \frac{5\pi}6; \frac{11\pi}6; \pi \right\} \)

1003085703

Část: 
A
Řešením rovnice \( 2\sin\!\left(x - \frac{\pi}6 \right) = 1 \), kde \( x\in\langle0; \pi\rangle \), je:
\( \left\{\frac{\pi}3; \pi \right\} \)
\( \left\{\frac{\pi}6 \right\} \)
\( \left\{\frac{\pi}3 \right\} \)
\( \left\{\frac{\pi}6; \frac{\pi}2 \right\} \)

1003085702

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

1003085701

Část: 
A
Najdi všechny \( x \), \( x\in\mathbb{R} \), pro které \( \mathrm{cotg}^2x = - \mathrm{cotg}\,x \).
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left[ \left\{\frac{3\pi}4+k\pi \right\}\cup\left\{\frac{\pi}2+k\pi \right\} \right] \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{3\pi}4+2k\pi \right\} \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{3\pi}4+k\pi \right\} \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}2+k\pi \right\} \)