sin x = (2 tg x/2)/(1+tg2x/2)
cos x = (1-tg2 2/x)/ (1+ tg² x/2)
sin2x = (2tgx)/(1+tg2x)
sin² a = 1/(1+ctg² a ) = tg² a /(1+tg² a )
cos² a = 1/(1+tg² a ) = ctg² a / (1+ctg² a )
ctg2a = (ctg² a -1)/ 2ctga
sin3a = 3sina -4sin³ a = 3cos² a sina -sin³ a
cos3a = 4cos³ a -3 cosa= cos³ a -3cosa sin² a
tg3a = (3tga -tg³ a )/(1-3tg² a )
ctg3a = (ctg³ a -3ctga )/(3ctg² a -1)
sin a /2 = ± Ö ((1-cosa )/2)
cos a /2 = ± Ö ((1+cosa )/2)
tga /2 = ± Ö ((1-cosa )/(1+cosa ))=
sina /(1+cosa )=(1-cosa )/sina
ctga /2 = ± Ö ((1+cosa )/(1-cosa ))=
sina /(1-cosa )= (1+cosa )/sina
sin(arcsin a ) = a
cos( arccos a ) = a
tg ( arctg a ) = a
ctg ( arcctg a ) = a
arcsin (sina ) = a ; a Î [-p /2 ; p /2]
arccos(cos a ) = a ; a Î [0 ; p ]
arctg (tg a ) = a ; a Î [-p /2 ; p /2]
arcctg (ctg a ) = a ; a Î [ 0 ; p ]
arcsin(sina )=
1)a - 2p k; a Î [-p /2 +2p k;p /2+2p k]
2) (2k+1)p - a ; a Î [p /2+2p k;3p /2+2p k]
arccos (cosa ) =
1) a -2p k ; a Î [2p k;(2k+1)p ]
2) 2p k-a ; a Î [(2k-1)p ; 2p k]
arctg(tga )= a -p k
a Î (-p /2 +p k;p /2+p k)
arcctg(ctga ) = a -p k
a Î (p k; (k+1)p )
arcsina = -arcsin (-a )= p /2-arccosa =
= arctg a /Ö (1-a ² )
arccosa = p -arccos(-a )=p /2-arcsin a =
= arc ctga /Ö (1-a ² )
arctga =-arctg(-a ) = p /2 -arcctga =
= arcsin a /Ö (1+a ² )
arc ctg a = p -arc cctg(-a ) =
= arc cos a /Ö (1-a ² )
arctg a = arc ctg1/a =
= arcsin a /Ö (1+a ² )= arccos1/Ö (1+a ² )
arcsin a + arccos = p /2
arcctg a + arctga = p /2