Goniometrické funkcie
Úloha 1
Priraď správnu funkčnú hodnotu:
A | B | C | D | E | F | G | H | ||
{\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}:
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 \}
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 \}
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 \}
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)
![](images/priklady/testn1k1.gif)
\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)
![](images/priklady/testn2k1.gif)
\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)
![](images/priklady/testn3k1.gif)
{\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} | ![]() |
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b) Funkcia sínus je pre x \in \mathbb{R} nepárna. | ![]() |
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c) Funkcia tangens je pre x \in \mathbb{R} neohraničená. | ![]() |
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d) Funkcia kotangens je pre x \in \mathbb{R} rastúca na celom definičnom obore. | ![]() |
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e) Funkcia kosínus je pre x \in \mathbb{R} neohraničená. | ![]() |
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f) \cos 180^\circ = -1 | ![]() |
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g) \rm tg {\large\pi \over 2} = 0 | ![]() |
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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 | ![]() |
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Úloha 5
Zisti predpis funkcie, ktorá je vykreslená na obrázku!
a)
![](images/priklady/sinustest.png)
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)
![](images/priklady/sinustest2.png)
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)
![](images/priklady/kosinustest.png)
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)
![](images/priklady/kosinustest2.png)
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)
![](images/priklady/tangenstest.png)
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)
![](images/priklady/tangenstest2.png)
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)
![](images/priklady/kotangenstest.png)
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)
![](images/priklady/kotangenstest2.png)
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
![](images/priklady/sintesta.png)
![](images/priklady/sintestb.png)
![](images/priklady/sintestc.png)
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
![](images/priklady/costesta.png)
![](images/priklady/costestb.png)
![](images/priklady/costestc.png)
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
![](images/priklady/tantestb.png)
![](images/priklady/tantestc.png)
![](images/priklady/tantesta.png)
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
![](images/priklady/cottestc.png)
![](images/priklady/cottestb.png)
![](images/priklady/cottesta.png)