Question

Fie x ∈ ( 0, π/2) si sin x=12/15. Sa se calculeze sin 2x, cos 2x, tg 2x, ctg 2x.

Answer 1

calculam cosx
din formula fundamentala (cosx)^2+(sinx)^2=1 => (cosx)^2=1-(sinx)^2 
(cosx)^2=1- (12/15)^2=1-144/225=225-144/225=81/225 => cosx=9/15 deoarece x∈(0,π/2) deci x>0 si de asta il lua pozitiv
sin2x=2sinxcosx
cos2x= are 3 formule 1. (cosx)^2-(sinx)^2=0
                                   2.1-2(sinx)^2=0
                                   3.2(cosx)^2-1=0
tg2x= 2tgx/1-(tgx)^2 sau poti s-o faci dupa tg2x= sin2x/cos2x
ctg2x=1/tg2x            sau                             ctg2x=cos2x/sin2x

rezolvarea:
 sin2x=2* 12/15 * 9/15=  216/15
cos2x=1-2* 144/225= 1-288/225=225-288/225=-63/15
tg2x=216/15 * 15/-63=72/-21
ctg2x=1/72/-21=-21/72