Identitati trigonometrice-clasa a-IX-aExercitii rezolvate sau cu indicatii.Formule utile:1) sin(x+y)=sinx cosy+siny cosx ;2) sin(x-y)=sinx cosy-siny cosx ;2) cos(x+y)=cosx cosy-sinx siny ; 4) cos(x-y)=cosx cosy+sinx siny.4) sin2x=2sinx cosx; 5) cos2x=cos2x-sin2x=2cos2x-1=1-2sin2x .Aplicatii:1)Aratati ca urmatoarele egalitati sunt adevarate: a) 1+sinx-cosx=22

,_

4 2cos2sin x x;Solutiie:2 2sin ,_

4 2cos2 x x= ,_

+4sin2sin4cos2cos2sin 2 2 x x x= ,_

+2sin2cos2sin 22 x x x=sinx+1-cosx =1+sinx-cosx ; sau: 1+sinx-cosx== ,_

+2cos2sin2sin 22x x x= ,_

+2cos2sin2sin 2x x x=1]1

,_

+2 2sin2sin2sin 2x x x = ,_

2 4cos2sin 2 2x x .b)1+cosx+cos2x=4 cosx cos ,_

,_

+2 6cos2 6x x ;Solutie:1+cosx+cos2x=2cos2x+cosx=2 cosx ,_

+21cos x=2 cosx ,_

+3cos cosx= = 4 cosx ,_

,_

+2 6cos2 6cosx x ;sau:

,_

,_

+2 6cos2 6cos cos 4x xx =1]1

,_

+ ,_

2sin6sin2cos6cos2sin6sin2cos6cos cos 4x x x xx ==1+cosx+cos2x.c) 1+sinx+cosx-tgx=xxxcos4cos2sin 2 22

,_

+Solutie:1+sinx+cosx-tgx=xx xx x xcos2cos2sin 22cos2sin 22sin 22 += ,_

+xx x xcos112cos2sin 22sin 22=( )xxxxxxxxxx xxxx xxxxx xxxx x x xcos4cos2sin 2 2cos4sin2sin 2 24cos4sincos2sin 4sin2sincos2sin 2sin coscos2sin 2cos2cos2sin 212sin 2cos1 cos2cos2sin 22sin 2222 22 2

,_

+

,_

,_

1]1

,_

,_

+ Exercitii selectate de prof.Iacovici Emiliad) sinx+siny+sin(x+y)=2cos2cos2sin 4y x y x +Solutie:sinx+siny+sin(x+y)=2cos2cos2sin 42cos2cos2sin 22cos2sin 22cos2sin 2y x y x y x y x y x y x y x y x y x + ,_

++ ++ ++ +Exercitii selectate de prof.Iacovici Emilia