Question

Calculați tg x/2, știind ca sinx+cosx=7/5 pentru x aprtine(0,π/4).​

Answer 1

Explicație pas cu pas:

fie:t=tg(x/2)

sinx=2t/(1+t²)

cosx=(1-t²)/(1+t²)

(-t²+2t+1)/(1+t²)=7/5

7+7t²=-5t²+10t+5

12t²-10t+2=0

6t²-5t+1=0

delta=25-24=1

t=(5+1)/12=1/2

t=(5-1)/12=1/3

x€(0,pi/4)

tg(x/2)€(0,tg(pi/8) )

tg(pi/8)=sin(pi/8)/cos(pi/8)

sin(pi/8)=sin(pi/4/2)=radical((1-rad2/2)/2)

radical((1-1,4/2)/2)=radical(0,15)=radical15/10=0,38

deci tg(x/2)€(0;0,38)

deci tg(x/2)=1/3

Raspuns:1/3

Bafta!