Bài 60 trang 219 SGK Đại số 10 Nâng cao

Đề bài

Nếu \(\sin \alpha  + \cos \alpha  = {1 \over 2}\) thì sin2α  bằng: 

\(\eqalign{
& (A)\,{3 \over 8}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(B)\, - {3 \over 4} \cr 
& (C)\,{1 \over {\sqrt 2 }}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(D)\,{3 \over 4} \cr} \)

Lời giải chi tiết

Ta có:

\(\eqalign{
& \sin \alpha + \cos \alpha = {1 \over 2}\cr & \Rightarrow {\left( {\sin \alpha  + \cos \alpha } \right)^2} = \frac{1}{4}\cr & \Rightarrow {\sin ^2}\alpha  + 2\sin \alpha \cos \alpha  + {\cos ^2}\alpha  = \frac{1}{4}\cr & \Rightarrow 1 + \sin 2\alpha = {1 \over 4} \cr 
& \Rightarrow \sin 2\alpha = - {3 \over 4} \cr} \)

Chọn (B)