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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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Solution re: proof found Comment 3 of 3 |
(In reply to proof found by Steven Lord)

Well, lets just skip the grunt work - deriving the formula for sin(5x) = poly in sin(x).

We can look that up; one place is https://mathworld.wolfram.com/Multiple-AngleFormulas.html .  {For brevity let S=sin(x)}  sin(5x) = 5S-20S^3+16S^5.

Now onto the actual problem: evaluate this at x=18 degrees.
Then 1 = 5S-20S^3+16S^5.  
Then subtract 1 from each side and factor: (S-1)*(4S^2+2S-1)^2 = 0.  
We know sin(18) = S != 1, so S must be one of the roots of 4S^2+2S-1 = 0.
The roots are S=(sqrt(5)-1)/4 and S=(-sqrt(5)-1)/4.  The second one is negative so it is rejected since we know that sin(18) is between 0 and 1.  Therefore sin(18) = (sqrt(5)-1)/4.

Now we can do what K Sengupta did and just directly evaluate the expression S^3+S^2 at (sqrt(5)-1)/4 to get 1/8.

  Posted by Brian Smith on 2023-09-18 10:49:24
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