(sin54)^(10) =x_1 x_(1 ) is root ax^2 +bx+c=0 a,b,c is integer a=?


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Question Number 182854 by manxsol last updated on 16/Dec/22

${(s i n 54)}^{10} = x_{1} x_{1} i s r o o t$ ${a x}^{2} + b x + c = 0$ $a, b, c i s i n t e g e r$ $a = ?$
Commented by manxsol last updated on 16/Dec/22

$o k, I w i l l c o r r e c t i t ., t h a n k s$ $f o r t h e s u g g e s t i o n .$ $H o w d i d ? y o u d o i t ?$
Commented by Frix last updated on 15/Dec/22

$? ? ?$ $a = 1$ $(x - x_{1}) (x - x_{2}) = x^{2} - (x_{1} + x_{2}) x + x_{1} x_{2}$ $b = - (x_{1} + x_{2})$ $c = x_{1} x_{2}$
Commented by manxsol last updated on 15/Dec/22

$s o r r y, i a m e d i t p r o b l e m$
Commented by manxsol last updated on 15/Dec/22

$t h a n k s$
Commented by Frix last updated on 15/Dec/22

$No need to be sorry .$ $\sin 54 ° = \frac{1 + \sqrt{5}}{4}$ $\sin^{10} 54 ° = \frac{123 + 55 \sqrt{5}}{2048}$
Commented by manxsol last updated on 15/Dec/22

$I s o l v e d i t l i k e t h i s$ $s i n 54 = \frac{\sqrt{5} + 1}{4} = \frac{1}{2} \frac{\sqrt{5} + 1}{2} = \frac{1}{2} Φ$ ${(\frac{1}{2})}^{10} Φ^{10} = \frac{1}{1024} (\frac{123 + 55 \sqrt{5}}{2})$ $= \frac{123 + 55 \sqrt{5}}{2048} = x_{1}$ $S e e {(★)}^{}$ $r o o t s i n t e g e r \Rightarrow x_{2 c o n j u g a t e}$ $x_{2} = \frac{123 - 55 \sqrt{5}}{2048}$ $x_{1} + x_{2} = \frac{246}{2048}$ $x_{1} x_{2} = \frac{4}{4194304}$ $c a l c u l a t e u s i n g e d i t o r t i n k u t a r a$ $2048^{2}$ $4194304.0$ $123^{2} - 5 \times 55^{2}$ $4.0$ $246 \times 2048$ $503808.0$ $4194304 / 4$ $1048576.0$ $503808 / 4$ $125952.0$ $i n p u t$ $4 \times 5$ $[p u s h] e n t e r$ $[p u s h] ⋮$ $[p u s h] c a l c u l a r$ $20.0$ $x^{2} - \frac{246}{2048} x + \frac{4}{4194304} = 0$ $4194304 x^{2} - 503808 x + 4 = 0$ $1048576 x^{2} - 125952 x + 1 = 0$ $s o l u t i o n : a = 1048576$ $(★)$ $Φ^{10} = {(\frac{1 + \sqrt{5}}{2})}^{10} = \frac{123 + 55 \sqrt{5}}{2}$ $s e r i e d e f i b o n a c c i$ $\begin{array}{cc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 55 & 89 \end{array}$ $a = 55$ $b = 123$ $d e d u c c i o n h e c h a d e t a b l a d e$ $p o t e n c i a s d e Φ = (1 + \sqrt{5}) / 2$
Commented by Frix last updated on 15/Dec/22

$You wrote a i s i n t e g e r .$ $You should have written a, b, c are integer .$
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