All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

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

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

Comments: ( Back to comment list | You must be logged in to post comments.)
 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°
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 54º-sin 18=sin18º*(2*(cos18º)^2t*cos36º-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

 Search: Search body:
Forums (0)