Question

sina + cosb= 1
cosa + sinb= -1/2
a,b apartin R
sin(a+b)=?​

Answer 1

Răspuns:

sin(a+b)=sin(a)cos(b)+cos(a)sin(b)

sin(a) + cos(b)= 1 ⇔(sin(a) + cos(b))²= 1² ⇔

sin(a)²+cos(b)²+2sin(a)cos(b)= 1  (1)

cos(a) + sin(b)= -1/2 ⇔[cos(a) + sin(b)]²= (-1/2)² ⇔

cos(a)² + sin(b)²+2cos(a)sin(b)=1/4 (2)

adunam relatiile (1) si (2) ⇒sin(a)²+cos(b)²+2sin(a)cos(b)+cos(a)² + sin(b)²+2cos(a)sin(b)=1+1/4 ⇔

[sin(a)²+cos(a)²]+[cos(b)²+sin(b)²] + 2[sin(a)cos(b)+cos(a)sin(b)]=1+1/4 ⇔

1+1+2sin(a+b)=1+1/4 ⇔2sin(a+b)=1/4-1 ⇔2sin(a+b) = -3/4

sin(a+b) = -3/8