Help

Question

Chứng minh

$\Delta ABC$ có ít nhất

$1$ góc bằng

$60$ nếu thỏa mãn:

$\frac{sinA+sinB+sinC}{cosA+cosB+cosC}=\sqrt{3}$

Kaito kid · Answer 1 · 8/8/2016 6:02:12 PM

$ta có \frac{sinA+sinB+sinC}{cosA+cosB+cosC}=\sqrt{3}$

$\Leftrightarrow sin A+sin B+sinC=\sqrt{3}(cosA+cosB+cosC)$

$\Leftrightarrow sinA-\sqrt{3}cosA+sinB-\sqrt{3}cosB+sinC-\sqrt{3}cosC=0$

$\Leftrightarrow \frac{sinA-\sqrt{3}cosA}{2}+\frac{sinB-\sqrt{3}cosB}{2}+\frac{sinC-\sqrt{3}cosC}{2}=0$

$\Leftrightarrow sin(A-\frac{\pi }{3})+sin(B-\frac{\pi}{3})+sin(C-\frac{\pi}{3})=0$

$\Leftrightarrow 2sin(\frac{A+B-\frac{2\pi}{3}}{2}).cos\frac{A-B}{2}+sin[\pi-(A+B)-\frac{\pi}{3}]$

$\Leftrightarrow 2sin(..........nt..............)+cos[\frac{\pi}{3}-(\frac{A+B}{2})]=0$

$\Leftrightarrow {2sin\frac{A+B-\frac{2\pi}{3}}{2}} =0(1)$ hoặc

$cos\frac{A+B}{2}=cos(\frac{\pi}{3}-\frac{A+B}{2})$ (2)

(1)

$\Rightarrow \widehat{C}=60$

(2)

$\Rightarrow \widehat{A}=\widehat{B}=60$

$\Rightarrow \Delta ABC$ có ít nhất 1 góc = 60

1 Đáp án