Goniometrické funkcie
Historické poznámky
Definície funkcií
Sínus ostrého uhla v pravouhlom trojuholníku je pomer dĺžky protiľahlej odvesny ostrého uhla k dĺžke prepony.
Kosínus ostrého uhla v pravouhlom trojuholníku je pomer dĺžky priľahlej odvesny ostrého uhla k dĺžke prepony.
applet
sin
sin
cos
cos![\alpha = \frac{b}{c} \alpha = \frac{b}{c}](https://lms.umb.sk/filter/tex/pix.php/defc51425bb3a9bdce94f253674e55fb.png)
Kosínus ostrého uhla v pravouhlom trojuholníku je pomer dĺžky priľahlej odvesny ostrého uhla k dĺžke prepony.
![](https://lms.umb.sk/pluginfile.php/438828/mod_book/chapter/11229/Koment%C3%A1r%202020-12-01%20122917.png)
![](https://lms.umb.sk/pluginfile.php/438828/mod_book/chapter/11229/Koment%C3%A1r%202020-12-01%20123847.png)
![( \alpha ) = ( \alpha ) =](https://lms.umb.sk/filter/tex/pix.php/0d534fe22f8f7d45f53072b7f72588a8.png)
![\alpha = \frac{a}{c} \alpha = \frac{a}{c}](https://lms.umb.sk/filter/tex/pix.php/28e0cd52afc46b70b52cd9695944e041.png)
![( \alpha ) = ( \alpha ) =](https://lms.umb.sk/filter/tex/pix.php/0d534fe22f8f7d45f53072b7f72588a8.png)
![\alpha = \frac{b}{c} \alpha = \frac{b}{c}](https://lms.umb.sk/filter/tex/pix.php/defc51425bb3a9bdce94f253674e55fb.png)
Tangens ostrého uhla v pravouhlom trojuholníku je pomer dĺžok protiľahlej a priľahlej odvesny ostrého uhla:
tg
tg
Kotangens ostrého uhla v pravouhlom trojuholníku je pomer dĺžok priľahlej a protiľahlej odvesny ostrého uhla:
cotg
cotg
tg
![( \alpha ) = ( \alpha ) =](https://lms.umb.sk/filter/tex/pix.php/0d534fe22f8f7d45f53072b7f72588a8.png)
![\alpha = \frac{a}{b} \alpha = \frac{a}{b}](https://lms.umb.sk/filter/tex/pix.php/a3316a6ed368783b3f2cc8f221b42f09.png)
Kotangens ostrého uhla v pravouhlom trojuholníku je pomer dĺžok priľahlej a protiľahlej odvesny ostrého uhla:
cotg
![( \alpha ) = ( \alpha ) =](https://lms.umb.sk/filter/tex/pix.php/0d534fe22f8f7d45f53072b7f72588a8.png)
![\alpha = \frac{b}{a} \alpha = \frac{b}{a}](https://lms.umb.sk/filter/tex/pix.php/2ad3c0a4e8095f352b341e9e988461fa.png)
Doplňte applet pre hodnoty funkcie tangens a kotangens.
Kosínus je pomer priľahlej odvesny k prepone. Priľahlá k uhlu
je odvesna
. Preto bude
cos
využijúc vzťah
90
dostaneme
sin
cos
cos (90
.
Vzťahy medzi sin a cos resp. tg a cotg
sin(
) = cos(90
); cos(
) = sin(90
)
tg(
) = cotg(90
); cotg(
) = tg(90
)
![\beta \beta](https://lms.umb.sk/filter/tex/pix.php/d12a0aadcfba3c64c44deb4d6bcc166a.png)
![a a](https://lms.umb.sk/filter/tex/pix.php/e49736f09a17efd3daec360132426f43.png)
cos
![\beta = \frac{a}{c} \beta = \frac{a}{c}](https://lms.umb.sk/filter/tex/pix.php/8df6bcaf71f37e4a6841b35e70ec1444.png)
využijúc vzťah
![\beta= \beta=](https://lms.umb.sk/filter/tex/pix.php/663c0a256e47e931320aa6bceedce3ae.png)
![^ \circ- \alpha ^ \circ- \alpha](https://lms.umb.sk/filter/tex/pix.php/bd618432487b1e0160fdca7f5ce21268.png)
sin
![\alpha = \frac{a}{c} = \alpha = \frac{a}{c} =](https://lms.umb.sk/filter/tex/pix.php/a21a9ab99605700071d4e4a0fbc49a32.png)
![\beta = \beta =](https://lms.umb.sk/filter/tex/pix.php/231cae3da89d95860ead79d86d95560e.png)
![^\circ- \alpha ) ^\circ- \alpha )](https://lms.umb.sk/filter/tex/pix.php/99cba2ef314bdba7f243655e09db2444.png)
Vzťahy medzi sin a cos resp. tg a cotg
sin(
![\alpha \alpha](https://lms.umb.sk/filter/tex/pix.php/a86a9f423465d727dd59fa89ba9cb8a5.png)
![^\circ- \alpha ^\circ- \alpha](https://lms.umb.sk/filter/tex/pix.php/c46b9e38bd79de17504666f2f7b4777d.png)
![\alpha \alpha](https://lms.umb.sk/filter/tex/pix.php/a86a9f423465d727dd59fa89ba9cb8a5.png)
![^\circ- \alpha ^\circ- \alpha](https://lms.umb.sk/filter/tex/pix.php/1ed1675b607a9d00b848dbc8b22a363e.png)
tg(
![\alpha \alpha](https://lms.umb.sk/filter/tex/pix.php/a86a9f423465d727dd59fa89ba9cb8a5.png)
![^\circ- \alpha ^\circ- \alpha](https://lms.umb.sk/filter/tex/pix.php/1ed1675b607a9d00b848dbc8b22a363e.png)
![\alpha \alpha](https://lms.umb.sk/filter/tex/pix.php/a86a9f423465d727dd59fa89ba9cb8a5.png)
![^\circ- \alpha ^\circ- \alpha](https://lms.umb.sk/filter/tex/pix.php/c46b9e38bd79de17504666f2f7b4777d.png)