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Golden sine (Posted on 2023-09-17) Difficulty: 2 of 5
Prove that sin318 + sin218 = 1/8.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 4.0000 (1 votes)

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proof found | Comment 1 of 3
Spoiler: hints ahead.....




The key step is finding a simple expression for sin 18. This is the essence of the problem. Once you derive an expression for sin 18, the rest is some obvious algebra. 

Accomplishing the first step directly (even from first principles) seemed for me improbable. But, I found a derivation and I give the barest outline of it here. This may help someone else to solve it: 

Start with noticing 5 x 18 = 90, and then use trig 
double and triple angle formulae (since 5 = 2 + 3). Next, a difference trig identity leads one to finally solve a quadratic, to gives one
 sin 18. 

This method for getting sin 18 may be found in full detail here


Edited on September 18, 2023, 3:08 am
  Posted by Steven Lord on 2023-09-17 16:28:13

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