Question

Fie x apartine (0,π/2),y apartine (π/2,π) cos x=1/5 sin y=2/3.Sa se calculeze cos (x-y) si ctg x

Answer 1

x∈(0;π/2)=>x∈ C 1=>sin x>0 ; cos x>0 (1)

y∈(π/2;π)=>y∈ C2=> sin y>0;cos y<0

sin²x+cos²x=1<=>sin²x+(1/5)²=1<=>sin²x+1/25=1<=>sin²x=1-1/25=>sin²x=24/25=>sin x=+-√(24/25)=+-2√6/5 (3)

Din (1) si (3) =>sin x=2√6/5

sin² y+cos²y=1<=>(2/3)²+cos²y=1<=>4/9+cos²y=1<=>cos²y=1-4/9=>cos²y=5/9=>cos y=+-√(5/9)=> cos y=+-√5/3 (4)

Din (2) si (4)=>cos y=-√5/3

cos(x-y)=cos x*cos y+sin x*sin y=1/5*(-√5/3)+2√6/5*2/3=-√5/15+4√6/15

ctg x= cos x/sin x=1/5/2√6/5=1/2√6=2√6/24=√6/12