Goniometrické funkcie
Vlastnosti sínus/kosínus
Definície funkcií sínus a kosínus pomocou jednotkovej kružnice zodpovedá definícii pre ostrý uhol v pravouhlom trojuholníku.
V pravouhlom trojuholníku
s ostrým uhlom
má protiľahlá odvesna veľkosť
, čo predstavuje -
-ovú súradnicu bodu
. Prepona má veľkosť
. Potom
sin
.
V pravouhlom trojuholníku
![SM_xM SM_xM](https://lms.umb.sk/filter/tex/pix.php/ac59aa839fab56573d643210514f0c7a.png)
![\alpha \alpha](https://lms.umb.sk/filter/tex/pix.php/a86a9f423465d727dd59fa89ba9cb8a5.png)
![|MM_x| |MM_x|](https://lms.umb.sk/filter/tex/pix.php/78083209862d65a66447c56017a2e600.png)
![y y](https://lms.umb.sk/filter/tex/pix.php/ee23c4f89cdc7b5be951059c2435fa2d.png)
![M M](https://lms.umb.sk/filter/tex/pix.php/30e1607d7260db1196cd907a6d5a280f.png)
![|SM| = 1 |SM| = 1](https://lms.umb.sk/filter/tex/pix.php/9bf3185c4a9fc4c9f5b1b73f11a60510.png)
sin
![(x) = y(M) (x) = y(M)](https://lms.umb.sk/filter/tex/pix.php/4026a22bbfcf1359d89406976d9f0fe0.png)
Symetria sin a cos
sin(π + x) = − sin(x), cos(π + x) = − cos(x)
sin(2π − x) = − sin(x), cos(2π − x) = cos(x)
Dôležité hodnoty pre 0° 30° 45° 60° 90°
sin(x)![\; \; 0 \;\; \;\frac{1}{2} \;\; \;\; \frac{ \sqrt {2}}{2} \;\; \; \frac{ \sqrt {3}}{2} \;\;\; 1 \; \; 0 \;\; \;\frac{1}{2} \;\; \;\; \frac{ \sqrt {2}}{2} \;\; \; \frac{ \sqrt {3}}{2} \;\;\; 1](https://lms.umb.sk/filter/tex/pix.php/3753004ba7549354553b777c8d1ac9f2.png)
cos(x)![\; 1 \;\; \; \frac{ \sqrt {3}}{2}\; \;\; \frac{ \sqrt {2}}{2} \;\; \;\frac{1}{2} \; \; \;\;0 \; 1 \;\; \; \frac{ \sqrt {3}}{2}\; \;\; \frac{ \sqrt {2}}{2} \;\; \;\frac{1}{2} \; \; \;\;0](https://lms.umb.sk/filter/tex/pix.php/b2e755de6b8700406451f5d832810bb0.png)
sin(π + x) = − sin(x), cos(π + x) = − cos(x)
sin(2π − x) = − sin(x), cos(2π − x) = cos(x)
Dôležité hodnoty pre 0° 30° 45° 60° 90°
sin(x)
![\; \; 0 \;\; \;\frac{1}{2} \;\; \;\; \frac{ \sqrt {2}}{2} \;\; \; \frac{ \sqrt {3}}{2} \;\;\; 1 \; \; 0 \;\; \;\frac{1}{2} \;\; \;\; \frac{ \sqrt {2}}{2} \;\; \; \frac{ \sqrt {3}}{2} \;\;\; 1](https://lms.umb.sk/filter/tex/pix.php/3753004ba7549354553b777c8d1ac9f2.png)
cos(x)
![\; 1 \;\; \; \frac{ \sqrt {3}}{2}\; \;\; \frac{ \sqrt {2}}{2} \;\; \;\frac{1}{2} \; \; \;\;0 \; 1 \;\; \; \frac{ \sqrt {3}}{2}\; \;\; \frac{ \sqrt {2}}{2} \;\; \;\frac{1}{2} \; \; \;\;0](https://lms.umb.sk/filter/tex/pix.php/b2e755de6b8700406451f5d832810bb0.png)
Autor: Daniel Mentrard. Dostupné Tu