Cho tam giác ABC. Khẳng định nào sau đây là đúng?

* Hướng dẫn giải

Tổng ba góc trong 1 tam giác bằng 180⁰  nên:

$$\begin{gathered} \hat{A}+\hat{B}+\hat{C}=180°\iff \hat{A}+\hat{B}=180°-\hat{C} \\ \iff \frac{\hat{A}+\hat{B}}{2}=90°-\frac{\hat{C}}{2}\Rightarrow cos\frac{A+B}{2}=\sin \frac{C}{2}; \sin \frac{A+B}{2}=cos\frac{C}{2} \end{gathered}$$

$$\begin{gathered} \sin A+\sin B+\sin C=2.\sin \frac{A+B}{2}.cos\frac{A-B}{2}+2\sin \frac{C}{2}.cos\frac{C}{2} \\ =2\cos \frac{C}{2}.\cos \frac{A-B}{2}+2\sin \frac{C}{2}.\cos \frac{C}{2}=2\cos \frac{C}{2}.\left(\cos \frac{A-B}{2}+\sin \frac{C}{2}\right) \\ =2\cos \frac{C}{2}.\left(\cos \frac{A-B}{2}+\cos \frac{A+B}{2}\right)=2\cos \frac{C}{2}.2\cos \frac{A}{2}.cos\frac{B}{2} \\ =4\cos \frac{A}{2}.cos\frac{B}{2}.\cos \frac{C}{2} \end{gathered}$$