Ta có với \(\forall \alpha \in \left( {0;\,\,\frac{\pi }{2}} \right) \Rightarrow \left\{ \begin{array}{l}\sin \alpha > 0\\\cos \alpha > 0\end{array} \right. \Rightarrow \left\{ \begin{array}{l}\sin \frac{\alpha }{2} > 0\\\cos \frac{\alpha }{2} > 0\end{array} \right..\)
\(a)\,\,\,A = \sin A + \sin B + \sin C\)
Ta có: \(0 < \,\,\angle A,\,\,\angle B,\,\,\angle C < \frac{\pi }{2} \Rightarrow \left\{ \begin{array}{l}\sin A > 0\\\sin B > 0\\\sin C > 0\end{array} \right. \Rightarrow A > 0.\)
\(b)\,\,\,B = \sin A.\sin B.\sin C\)
Tương tự câu a) ta có: \(B > 0.\)
\(c)\,\,C = \cos \frac{A}{2}.\cos \frac{B}{2}.\cos \frac{C}{2}\)
Ta có: \(0 < \,\,\angle A,\,\,\angle B,\,\,\angle C < \frac{\pi }{2} \Rightarrow 0 < \angle A,\,\,\angle B,\,\,\angle C < \frac{\pi }{4} \)
\( \Rightarrow \left\{ \begin{array}{l}\cos \frac{A}{2} > 0\\\cos \frac{B}{2} > 0\\\cos \frac{C}{2} > 0\end{array} \right. \Rightarrow C > 0.\)
\(d)\,\,D = \tan \frac{A}{2} + \tan \frac{B}{2} + \tan \frac{C}{2}\)
Ta có: \(0 < \,\,\angle A,\,\,\angle B,\,\,\angle C < \frac{\pi }{2} \Rightarrow 0 < \angle A,\,\,\angle B,\,\,\angle C < \frac{\pi }{4} \)
\( \Rightarrow \left\{ \begin{array}{l}\tan \frac{A}{2} > 0\\\tan \frac{B}{2} > 0\\\tan \frac{C}{2} > 0\end{array} \right. \Rightarrow D > 0.\)
