Bài 111007

Question

Cho $\Delta ABC$ có $\widehat{C}=2 \widehat{B}=4 \widehat{A} $
a) Tính $\widehat{A}, \widehat{B}, \widehat{C} $ b) Chứng minh rằng $\frac{1}{a}=\frac{1}{b}+\frac{1}{c} $.

Kit Nguyen · Answer 1 · 7/7/2012 9:52:01 AM

a) Ta có: $A+B+C= \pi    \Leftrightarrow    A+2A+4A=7A=\pi$
$\Rightarrow     A=\frac{\pi}{7}; B=2A=\frac{2 \pi}{7}; C=4A=\frac{4\pi}{7}.   $

b) Ta có: $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}    \Leftrightarrow     bc=ab+ac$
Vì   $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R   $
$\Rightarrow    a=2R \sin A; b=2R \sin B; c=2R \sin C$
Nên $bc=ab+ac$
$\Leftrightarrow    \sin \frac{2\pi}{7}\sin \frac{4 \pi}{7}=\sin \frac{\pi}{7} \sin \frac{2 \pi}{7}+\sin \frac{\pi}{7} \sin \frac{4 \pi}{7}      $
$\Leftrightarrow    2\sin \frac{2\pi}{7}\sin \frac{4 \pi}{7}=\cos \frac{\pi}{7}-\cos \frac{3 \pi}{7}+\cos \frac{3 \pi}{7}-\cos \frac{5 \pi}{7}      $
$\Leftrightarrow    2\sin \frac{2\pi}{7}\sin \frac{4 \pi}{7}=\cos \frac{\pi}{7}-\cos \frac{5 \pi}{7}    $
$\Leftrightarrow    2\sin \frac{4\pi}{7}\sin \frac{2\pi}{7}=2 \sin \frac{3 \pi}{7}\sin \frac{2 \pi}{7}    $
$\Leftrightarrow    \sin \frac{4 \pi}{7}=\sin \frac{3\pi}{7}   $ đúng   (vì $\frac{4\pi}{7}+\frac{3\pi}{7}=\pi).   $

Bài 111007

1 Đáp án

Lý thuyết liên quan

Bài 111007

1 Đáp án

Liên quan

Lý thuyết liên quan