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Not So Simple Sines (Posted on 2009-12-24) Difficulty: 3 of 5
Analytically prove sin(54)-sin(18) = 1/2

See The Solution Submitted by Brian Smith    
Rating: 4.0000 (2 votes)

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Solution not so complicated | Comment 1 of 4

 

It is easy to derIve the irrational expression of sin18:

Starting with identity
 sin 72 = 2 sin 36 cos 36                                 thru
cos 18 = 2 sin 36 cos 36                               
  we arrive to quadratic equation
with one relevant root
t= 1/4*(sqrt5-1)                                                       and    t^2=1/8* (-sqrt5+3)         
where            t=  sin 18

Now, back to our problem:

Applying few trigonmetric trivial identities ,we get

Z=sin 54-sin 18=sin18*(2*(cos18)^2t*cos36-1)=
=t*(2-2t^2+1)

Z=1/4*(sqrt5-1)*(2-4*1/8* (-sqrt5+3) )=
2*Z=(sqrt5-1)*(sqrt5+1)/4=(5-1) /4=1

Z=1/2   QED     


  Posted by Ady TZIDON on 2009-12-25 05:33:19
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