Let-x-cos-pi-9-Show-that-8x-3-6x-1-0-Deduce-x-is-not-rational-

Question Number 207194 by sniper237 last updated on 09/May/24

L e t x = c o s \frac{π}{9}

S h o w t h a t 8 x^{3} - 6 x - 1 = 0

D e d u c e x i s n o t r a t i o n a l

Answered by Berbere last updated on 09/May/24

c o s (3 . \frac{π}{9}) = c o s (\frac{π}{3}) = \frac{1}{2}

c o s (3 a) = R e {(e^{i a})}^{3} = 4 {c o s}^{3} (a) - 3 c o s (a)

\Rightarrow \frac{1}{2} = 4 x^{3} - 3 x \Rightarrow 8 x^{3} - 6 x - 1 = 0

s u u p o s e x = \frac{p}{q}; g c d (p, q) = 1;

\Rightarrow 8 p^{3} - 6 {p q}^{2} - q^{3} = 0 \Leftrightarrow p (8 p^{2} - 6 q^{2}) = q^{3}

\Rightarrow p ∣ q^{3} \Rightarrow (g c d (p, q) = 1 \Rightarrow g c d (p, q^{3}) = 1 \Rightarrow p = 1

\Rightarrow 8 - 6 q^{2} - q^{3} = 0 \Rightarrow q^{2} ∣ 8 \Rightarrow q \in {1, 2} x = 1; x = \frac{1}{2} i m p o s s i b l e

\Rightarrow x \notin IQ

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