Function sinus, cosinus

Definition:

   
Let x be individual real number and M[xm, ym] point in unitary circle constructed in kartezian coordinate system, which is given to number x in ilustration U.
Function sinus is function, which gives to every x Î R ym.
Funkcia cosinus is function, which gives to every x Î R priraïuje xm.


Definition field of function sinus and cosinus is set R.
In table are shown values of functions sinus a cosinus in several points of interval <0, 2p).


x
  0  
= 30°
= 45°
= 60°
= 90°
p = 180°
= 270°
sin x
  0  
1
0
-1
cos x
  1  
0
-1
0


Both function are periodic with period 2p, which is also their smalest period.

Sentence:

For every k Î Z and for every x Î R is
sin(x + k.2p) = sin x
cos(x + k.2p) = cos x
We see (unitary circle) that
For every x Î <0, 2p) is |cos x| £ 1 a |sin x| £ 1. Both function are according to that in interval <0, 2p) down and up limited.

Function y = sin x:
Maximum in x1 = p/2 on interval <0, 2p)
Minimum in x2 = 3/2p on interval <0, 2p).
sin (-x) = -sin x => uneven
graf = sínusoid




Function y = cos x

Maximum in x3 = 0 on interval <0, 2p)
Minimum in x4 = p on interval <0, 2p).
cos (-x) = cos x => even
graf = cosínusoid



»
sinus
increasing
decreasing
decreasing
increasing
cosinus
decreasing
decreasing
increasing
increasing


»
sinus
+
+
-
-
cosinus
+
-
-
+
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