Goniometrické funkcie

Úloha 1

Priraď správnu funkčnú hodnotu:

ABC DEFGH
{\sqrt{3} \over 3} {\sqrt{3} \over 2} {\sqrt{2} \over 2} -1 0 {1 \over 2} 1 \sqrt{3}
a) \sin \large {\pi \over 6}
b) \cos \large {\pi \over 4}
c) {\rm tg}\: \large {\pi \over 3}
d) {\rm cotg}\: \large {\pi \over 3}
e) \sin \large {3\pi \over 2}
f) \cos \large {\pi \over 6}
g) {\rm tg}\: \pi
h) {\rm cotg}\: \large {\pi \over 4}

Úloha 2

Urči riešenie danej rovnice s neznámou x \in \mathbb{R}:

a)\cos x = -1

K = \bigcup\limits_{k\in\mathbb{Z}} \left \{\ 0 + 2k\pi \right \}
K = \bigcup\limits_{k\in\mathbb{Z}} \left \{\ -{\pi \over 2} + 2k\pi \right \}
K = \bigcup\limits_{k\in\mathbb{Z}} \left \{\ -\pi + 2k\pi \right \}

b)\sin x = {\sqrt{3} \over 2}

K = \bigcup\limits_{k\in\mathbb{Z}} \left \{\ {\pi \over 6} + 2k\pi; {5 \over 6}\pi + 2k\pi \right \}
K = \bigcup\limits_{k\in\mathbb{Z}} \left \{\ {\pi \over 3} + 2k\pi; {2 \over 3}\pi + 2k\pi \right \}
K = \bigcup\limits_{k\in\mathbb{Z}} \left \{\ {\pi \over 4} + 2k\pi; {3 \over 4}\pi + 2k\pi \right \}

c){\rm tg}\: x = 1

K = \bigcup\limits_{k\in\mathbb{Z}} \left \{\ 2k\pi \right \}
K = \bigcup\limits_{k\in\mathbb{Z}} \left \{\ {\pi \over 2} + k\pi \right \}
K = \bigcup\limits_{k\in\mathbb{Z}} \left \{\ {\pi \over 4} + k\pi \right \}

d){\rm cotg}\: x = -1

K = \bigcup\limits_{k\in\mathbb{Z}} \left \{\ {3 \over 4}\pi + 2k\pi \right \}
K = \bigcup\limits_{k\in\mathbb{Z}} \left \{\ -{\pi \over 4} + k\pi \right \}
K = \bigcup\limits_{k\in\mathbb{Z}} \left \{\ {\pi \over 4} + k\pi \right \} Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 207

Úloha 3

Zisti akému zadaniu odpoveda červenou farbou vyznačené riešenie na jednotkovej kružnici.

a)


\sin x \geq {1 \over 2}
\cos x > {1 \over 2}
\sin x > {1 \over 2} Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 221

b)


\sin x \leq 0
\cos x > 0
\cos x < 0 Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 235

c)


\cos x \leq 1
{\rm tg}\: x \leq 1
{\rm cotg}\: x \leq 1

Úloha 4

Rozhodni, či dané tvrdenie je pravdivé:

a) \sin {\pi \over 6} = {1 \over 2}
ano
nie
b) Funkcia sínus je pre x \in \mathbb{R} nepárna.
ano
nie
c) Funkcia tangens je pre x \in \mathbb{R} neohraničená.
ano
nie
d) Funkcia kotangens je pre x \in \mathbb{R} rastúca na celom definičnom obore.
ano
nie
e) Funkcia kosínus je pre x \in \mathbb{R} neohraničená.
ano
nie
f) \cos 180^\circ = -1
ano
nie
g) \rm tg {\large\pi \over 2} = 0
ano
nie
h) {\rm cotg}\: x = \large{\frac {\sin x} {\cos x}} Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 363
ano
nie

Úloha 5

Zisti predpis funkcie, ktorá je vykreslená na obrázku!

a)


y = 2{\rm sin}\: x - 1
y = 2{\rm cos}\: x - 1
y = 2{\rm cos}\: x Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 377

b)


y = |{\rm sin}\: x | - \frac {1}{2}
y = |{\rm sin}\: x - 1 |
y = |{\rm sin}\: x - \frac {1}{2} | Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 391

c)


y = {\rm cos}\: (x + \frac {\pi}{6})
y = {\rm cos}\: (x + \frac {\pi}{4})
y = {\rm cos}\: (x + \frac {\pi}{3}) Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 405

d)


y = {\rm cos}\: (x + \frac {\pi}{4}) - 1
y = {\rm cos}\: (x + \frac {\pi}{3}) - 1
y = {\rm cos}\: (x + \frac {\pi}{6}) - 1 Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 419

e)


y = 2{\rm tg}\: x - 1
y = {\rm tg}\: x - 1
y = 2{\rm tg}\: x Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 433

f)


y = {\rm tg}\: (x + \frac {\pi}{3})
y = {\rm tg}\: (x + \frac {\pi}{4})+ 1
y = {\rm tg}\: (x + \frac {\pi}{3})+ 1 Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 447

g)


y = 1 - {\rm cotg}\: x
y = 2 - {\rm cotg}\: x
y = {\rm cotg}\: x Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 461

h)


y = |{\rm cotg}\: x - 1 |
y = |{\rm cotg}\: x + 1 |
y = 2|{\rm cotg}\: x - 1 |

Úloha 6

Zisti, ktorý graf odpovedá zadanému predpisu funkcie!

a)

f(x) = |{\rm sin}\: x + \frac {\pi}{6} | Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 479 Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 480 Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 481





b)

f(x) = 2{\rm cos}\: x + \frac {\pi}{4} Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 493 Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 494 Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 495





c)

f(x) = \frac {1}{2} {\rm tg}\: x + 1 Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 507 Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 508 Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 509





d)

f(x) = 2{\rm cotg}\: x -1 Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 521 Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 522 Notice: Undefined variable: prefix in /srv/beegfs/web/web/kdm/diplomky/matus_kepic_dp/gon_rovnice_nerovnice/include/testy.gonio.inc on line 523