Khi sinA = (cosB + cosC)/(sinB + sinC) thì tam giác ABC là tam giác gì? Tam giác đều

* Hướng dẫn giải

Đáp án C

Ta có:

$$\begin{gathered} \frac{\cos B+\cos C}{\sin B+\sin C}=\frac{2\cos \frac{B+C}{2}.\cos \frac{B-C}{2}}{2\sin \frac{B+C}{2}\cos \frac{B-C}{2}} \\ =\frac{\cos \frac{B+C}{2}}{\sin \frac{B-C}{2}}=\frac{\cos \left(\frac{\pi }{2}-\frac{A}{2}\right)}{\sin \left(\frac{\pi }{2}-\frac{A}{2}\right)} \\ =\frac{\sin \frac{A}{2}}{\cos \frac{A}{2}}\Rightarrow \sin A=\frac{\sin \frac{A}{2}}{\cos \frac{A}{2}} \\ \Rightarrow 2\sin \frac{A}{2}\cos \frac{A}{2}=\frac{\sin \frac{A}{2}}{\cos \frac{A}{2}} \\ \Rightarrow 2{\cos }^2\frac{A}{2}=1\Rightarrow \cos A=0\Rightarrow A={90}^0 \end{gathered}$$