Mathc initiation/a007
Vérifier quelques propriétés mathématiques de trigonométrie
3tan(x) -tan(x)**3 Vérifions si : tan(3x ) = ------------------ 1-3tan(x)**2
Nous avons vu que :
tan(y) + tan(x)
tan(y+x) = ----------------
1 - tan(y)tan(x)
posons y = 2x :
tan(2x) + tan(x)
tan(2x+x) = ----------------
1 - tan(2x)tan(x)
tan(2x) + tan(x) (a)
tan(3x) = ----------------
1 - tan(2x)tan(x) (b)
a) ------------------------------------------
2tan(x)
tan(2x) + tan(x) = ----------- + tan(x)
1-tan(x)**2
2tan(x) tan(x) (1-tan(x)**2)
= ----------- + ---------------------
1-tan(x)**2 1-tan(x)**2
2tan(x) tan(x)-tan(x)**3
= ----------- + ----------------
1-tan(x)**2 1-tan(x)**2
3tan(x) -tan(x)**3
= --------------------
1-tan(x)**2
b) ------------------------------------------
2tan(x)
1 - tan(2x)tan(x) = 1 - ----------- tan(x)
1-tan(x)**2
1-tan(x)**2 2tan(x)**2
= ----------- - -----------
1-tan(x)**2 1-tan(x)**2
1-3tan(x)**2
= ------------
1-tan(x)**2
a/b) ------------------------------------------
3tan(x) -tan(x)**3
------------------
1-tan(x)**2
tan(3x ) = ------------------------
1-3tan(x)**2
------------
1-tan(x)**2
donc
3tan(x) -tan(x)**3
tan(3x ) = ------------------
1-3tan(x)**2