Question

sin x = 9/15 x apartine (pi/2 ;pi)
sin y = 12/15 y apartine (pi; 3pi/2)
cos (x+y)=? sin(x+3pi/4)=?

Vreo idee cineva? Multumesc.

Answer 1

Răspuns:

Explicație pas cu pas:

sinx=9/15, din sin²x+cos²x=1, ⇒(9/15)²+cos²x=1, ⇒cos²x=1-(9/15)²=(1-(9/15))(1+(9/15))=(6/15)·(24/15)=(6²·4)/(15²). Deci cosx=-12/15, deoarece in cadr.2 cosx<0.

y ∈cadr.3, deci siny=-12/15 si cosy<0

Din  sin²y+cos²y=1, ⇒(-12/15)²+cos²x=1, ⇒cos²x=1-(12/15)²=(1-(12/15))(1+(12/15))=(3/15)·(27/15)=(3²·9)/(15²). Deci cosy=-9/15,

cos(x+y)=cosx·cosy-sinx·siny=(-12/15)·(-9/15)-(9/15)·(-12/15)=2·(12/15)·(9/15)=(2·12·9)/(15·15)=24/25.

sin(x+3pi/4)=sinx·cos(3pi/4)+cosx·sin(3pi/4)=(9/15)·cos(π- π/4)+(-12/15)·sin(π- π/4)=(9/15)·(-cos(π/4))+(-12/15)·sin(π/4)=(9/15)·(-√2/2)- (12/15)·(√2/2)=-(√2/2)·((9/15)+(12/15))=-(√2/2)·(17/15)=-17√2/30.