If-cyc-sin-A-pi-6-cos-B-pi-6-cos-C-pi-6-in-ABC-Solve-for-real-numbers-x-4-4-x-3-6-x-2-4-x-1-0-

Question Number 188226 by Shrinava last updated on 26/Feb/23

If Ω = \sum_{cyc} \frac{\sin (A - \frac{π}{6})}{\cos (B - \frac{π}{6}) \cos (C - \frac{π}{6})} in △ ABC

Solve for real numbers :

x^{4} - 4 Ω x^{3} + 6 Ω x^{2} - 4 Ω x + 1 = 0

Answered by aleks041103 last updated on 27/Feb/23

s u p p o s e A B C e q u i l a t e r a l

A = B = C = \frac{π}{3}

\Rightarrow Ω = \frac{3 s i n (π / 6)}{{c o s}^{2} (π / 6)} = 3 \frac{\frac{1}{2}}{{(\frac{\sqrt{3}}{2})}^{2}} = \frac{3}{2} \frac{4}{3} = 2

\Rightarrow x^{4} - 8 x^{3} + 12 x^{2} - 8 x + 1 = 0

x = 0 i s n o t s o l

\Rightarrow x^{2} + \frac{1}{x^{2}} - 8 (x + \frac{1}{x}) + 12 = 0

y = x + \frac{1}{x} \Rightarrow y^{2} = x^{2} + \frac{1}{x^{2}} + 2

\Rightarrow y^{2} - 2 - 8 y + 12 = 0

y^{2} - 8 y + 10 = 0

y_{1, 2} = \frac{8 \pm \sqrt{64 - 4.10}}{2} = \frac{8 \pm 2 \sqrt{6}}{2} = 4 \pm \sqrt{6} \approx 1.55, 6.45

x^{2} - y x + 1 = 0

\Rightarrow x_{1, 2, 3, 4} = \frac{y_{1, 2} \pm \sqrt{y_{1, 2}^{2} - 4}}{2} = \frac{y_{1, 2} \pm \sqrt{6 - 8 y_{1, 2}}}{2}

b u t 6 - 8 y_{1, 2} < 0 \Rightarrow x \notin R

Commented by Shrinava last updated on 28/Feb/23

perfect dear professor thank you