Question-65723 – Tinku Tara

Question Number 65723 by bshahid010@gmail.com last updated on 02/Aug/19

Answered by Tanmay chaudhury last updated on 03/Aug/19

7 a = 2 π

4 a = 2 π - 3 a

c o s (4 a) = c o s (2 π - 3 a)

(2 {c o s}^{2} 2 a - 1) = c o s 3 π

c o s a = c s o c o s 2 a = 2 c^{2} - 1

c o s 3 a = 4 c^{3} - 3 c

{2 {(2 c^{2} - 1)}^{2} - 1} = (4 c^{3} - 3 c)

{2 (4 c^{4} - 4 c^{2} + 1) - 1} = (4 c^{3} - 3 c)

8 c^{4} - 8 c^{2} + 1 = 4 c^{3} - 3 c

8 c^{4} - 4 c^{3} - 8 c^{2} + 3 c + 1 = 0

8 c^{4} - 8 c^{3} + 4 c^{3} - 4 c^{2} - 4 c^{2} + 4 c - c + 1 = 0

8 c^{3} (c - 1) + 4 c^{2} (c - 1) - 4 c (c - 1) - 1 (c - 1) = 0

(c - 1) (8 c^{3} + 4 c^{2} - 4 c - 1) = 0

8 c^{3} + 4 c^{2} - 4 c - 1 = 0

c = c o s a = c o s (\frac{2 π}{7}) [s i n c e 7 a = 2 π]

c o s (\frac{2 π}{7}) i s a r o o t o f

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

Commented by bshahid010@gmail.com last updated on 03/Aug/19

thank you

Commented by Tanmay chaudhury last updated on 03/Aug/19

m o s t w e l c o m e

Commented by MJS last updated on 03/Aug/19

but \cos \frac{π}{7} d o e s solve the equation \dots

the other roots are \cos \frac{3 π}{7} and \cos \frac{5 π}{7}

8 (x - \cos \frac{π}{7}) (x - \cos \frac{3 π}{7}) (x - \cos \frac{5 π}{7}) \approx

\approx 8 (x - . 900 968 868) (x - . 222 520 934) (x + . 623 489 801) \approx

\approx 8 x^{3} - 4 x^{2} - 4 x + 1

Commented by MJS last updated on 05/Aug/19

I still {don}^{'} t understand why you did what you

did . it works exactly the sane way with the

given value

7 a = π

4 a = π - 3 a

\cos 4 a = - \cos 3 a

a = \arccos c

8 c^{4} - 8 c^{2} + 1 = 3 c - 4 c^{3}

8 c^{4} + 4 c^{3} - 8 c^{2} - 3 c + 1 = 0

(c + 1) (8 c^{3} - 4 c^{2} - 4 c + 1) = 0

a = \frac{π}{7} \Rightarrow \cos \frac{π}{7} is a root of the given equation

Commented by Tanmay chaudhury last updated on 05/Aug/19

s i r y o u r a n s w e r a l s o c o r r e c t \dots y o u h a v e t a k e n

7 a = π a n d i h a v e t a k e n 7 a = 2 π

b o t h a r e c o r r e c t \dots e q u a t i o n a r e d i f f e r e n t \dots

i t h o u g h t 7 a = 2 π s o i d i d i t \dots

y o u r a n z w e r i s a s p e r g i v v e n c o n d i t i o n i n a u e s t i o n

g