Question

cho α , β thỏa mãn sin α + sin β =\(\frac{\sqrt{2}}{2}\) và cos α + cos β =\(\frac{\sqrt{6}}{2}\).Tính sin( α + β )

Answer 1

\(sina+sinb=2sin\left(\frac{a+b}{2}\right)cos\left(\frac{a-b}{2}\right)=\frac{\sqrt{2}}{2}\)

\(\Rightarrow sin\left(\frac{a+b}{2}\right)cos\left(\frac{a-b}{2}\right)=\frac{\sqrt{2}}{4}\) (1)

\(cosa+cosb=2cos\left(\frac{a+b}{2}\right)cos\left(\frac{a-b}{2}\right)=\frac{\sqrt{6}}{2}\)

\(\Rightarrow cos\left(\frac{a+b}{2}\right)cos\left(\frac{a-b}{2}\right)=\frac{\sqrt{6}}{4}\) (2)

(1); (2) \(\Rightarrow tan\left(\frac{a+b}{2}\right)=\frac{\sqrt{3}}{3}\) \(\Rightarrow tan\left(a+b\right)=\sqrt{3}\) \(\Rightarrow a+b=60^0\)

\(\Rightarrow sin\left(a+b\right)=sin\left(60^0\right)=\frac{\sqrt{3}}{2}\)