Solve sin &#x2061;<!-- ⁡ --> ( &#x03C0;<!-- π --> 5 </mfrac> ) analy

Boilanubjaini8f

Boilanubjaini8f

Answered question

2022-06-27

Solve sin ( π 5 ) analytically
first step I have to
Show that: cos ( π 5 ) sin ( π 10 ) = 1 2
My question is, why do I have to do that?

Answer & Explanation

nuvolor8

nuvolor8

Beginner2022-06-28Added 32 answers

A simple way is provided by a geometric approach, rather than an analytic one.
It is well-known that if ABCDE is a regular pentagon, A C A B = φ = 1 + 5 2
Since A B C ^ = 3 π 5 , that implies sin 3 π 10 = φ 2 and
cos 3 π 10 = sin π 5 = 1 ( φ 2 ) 2 = 5 5 8 .
landdenaw

landdenaw

Beginner2022-06-29Added 8 answers

By repeated application of angle sum formulas we may get,
sin ( 5 x ) = sin 5 x + 5 cos 4 x sin x 10 sin 3 x cos 2 x
Let x = π 5 and let sin ( π 5 ) = u then we have,
0 = u 5 + 5 ( 1 u 2 ) 2 u 10 ( 1 u 2 ) u 3
It is safe to say 2 2 > u > 0. So that we may divide by u to get.
0 = u 4 + 5 ( 1 u 2 ) 2 10 ( 1 u 2 ) u 2
0 = 16 u 4 20 u 2 + 5
By solving this for u 2 first and then u you get the only root in ( 0 , 2 2 ) to be,
u = 1 2 5 2 5 2

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