Cho tam giác ABC có các góc thỏa mãn: \(2c{\rm{osA + cosB + cosC = }}\frac{9}{4}\).

Tính \(\sin \frac{A}{2}\).

Lời giải

Ta có:

\(2c{\rm{osA + cosB + cosC = }}\frac{9}{4}\)

\( \Leftrightarrow 2c{\rm{osA + 2cos}}\left( {\frac{{{\rm{B + C}}}}{2}} \right){\rm{cos}}\left( {\frac{{B - C}}{2}} \right){\rm{ = }}\frac{9}{4}\)

\( \Leftrightarrow c{\rm{osA + cos}}\left( {\frac{{{\rm{B + C}}}}{2}} \right){\rm{cos}}\left( {\frac{{B - C}}{2}} \right){\rm{ = }}\frac{9}{8}\)

\( \Leftrightarrow c{\rm{osA + sin}}\frac{A}{2}{\rm{cos}}\left( {\frac{{B - C}}{2}} \right){\rm{ = }}\frac{9}{8}\)

\( \Leftrightarrow 1 - 2{\sin ^2}\frac{A}{2}{\rm{ + }}\sin A{\rm{cos}}\left( {\frac{{B - C}}{2}} \right){\rm{ = }}\frac{9}{8}\)

\( \Leftrightarrow - 2{\sin ^2}\frac{A}{2}{\rm{ + }}\sin \frac{A}{2}{\rm{cos}}\left( {\frac{{B - C}}{2}} \right) - \frac{1}{8}{\rm{ = 0}}\)                          (1)

Để tồn tại góc A thì phương trình (1) phải có nghiệm \(\sin \frac{A}{2}\)

Suy ra ∆ ≥ 0

\( \Leftrightarrow co{s^2}\left( {\frac{{B - C}}{2}} \right) - 1 \ge 0\)

\( \Leftrightarrow cos\left( {\frac{{B - C}}{2}} \right) = 1\)

Khi đó \( - 2{\sin ^2}\frac{A}{2}{\rm{ + }}\sin \frac{A}{2} - \frac{1}{8}{\rm{ = 0}}\)

\( \Leftrightarrow {\sin ^2}\frac{A}{2}{\rm{ }} - \frac{1}{2}\sin \frac{A}{2} + \frac{1}{{16}}{\rm{ = 0}}\)

\( \Leftrightarrow {\left( {\sin \frac{A}{2} - \frac{1}{4}} \right)^2}{\rm{ = 0}}\)

\( \Leftrightarrow \sin \frac{A}{2} = \frac{1}{4}\)

Vậy \(\sin \frac{A}{2} = \frac{1}{4}\).