miércoles, 11 de septiembre de 2024

Golden Number, angle 36, cos(36°)=(1+√5)/4=Phi/2=GoldenNumber/2

Golden Number, angle 36, cos(36°)=(1+√5)/4=Phi/2=GoldenNumber/2

https://www.cut-the-knot.org/pythagoras/cos36.shtml 



cos 36°

The purpose of this page is to establish that

cos(36)=(1+5)4.

It's a good exercise in trigonometry that was also useful in solving a curious sangaku problem.

We start with a regular pentagon. As every regular polygon, this one too is cyclic. So that we may assume its vertices lie on a circle and sides and diagonals form inscribed angles.

It follows that every angle of the regular pentagon equals 108. Inscribed CAD is half of the central angle 72=3605, i.e.

CAD=36.

By symmetry BAC=DAE implying that these two are also 36. (Note in passing that we just proved that angle 108 is trisectable.) Other angles designated in the diagram are also easily calculated.

As far as linear segments are concerned, we may observe that triangles ABPABEAEP are isosceles, in particular,

AB=BP,
AB=AE,
AP=EP.

Triangles ABE and AEP are also similar, in particular

BE/AB=AE/EP, or
BE×EP=AB2, i.e.
(BP+EP)×EP=AB2, and lastly,
(AB+EP)×EP=AB2.

For the ratio x=AB/EP we have the equation

x+1=x2,

with one positive solution x=ϕ, the golden ratio. (These calculations were in fact performed to justify a construction of regular pentagon. We return to them here in order to facilitate the references.)

In AEPAE=AB and EP is one of the sides such that AE/EP=ϕ. Drop a perpendicular from P to AE to obtain two right triangles. Then say,

cos(AEP)=(AE/2)/EP=(AE/EP)/2=ϕ/2.

But AEP=36 and we get the desired result.

Using cos(36)=(1+5)/4 we can find

cos(18)=2(5+5)/4

from cos2α=2cos2α1 and then

sin(18)=2(35)/4

from cos2α+sin2α=1. Now, it may be hard to believe but this expression simplifies to

sin(18)=(51)/4,

which is immediately verified by squaring the two expressions. Also

sin(36)=2(55)/4

from sin2α=2 sinα cosα.

We can easily find cos(72) from cos2α=2cos2α1:

cos(72)

This is of course equal to sin(18) as might have been expected from the general formula, sinα=cos(90α).

And, of course,

sin(72)=2(5+5)/4

because sin(72)=cos(18).

Finally, let's compute sin54:

sin54

as expected since sin54=cos36.

Trigonometry

Fibonacci Numbers

  1. Ceva's Theorem: A Matter of Appreciation
  2. When the Counting Gets Tough, the Tough Count on Mathematics
  3. I. Sharygin's Problem of Criminal Ministers
  4. Single Pile Games
  5. Take-Away Games
  6. Number 8 Is Interesting
  7. Curry's Paradox
  8. A Problem in Checker-Jumping
  9. Fibonacci's Quickies
  10. Fibonacci Numbers in Equilateral Triangle
  11. Binet's Formula by Inducion
  12. Binet's Formula via Generating Functions
  13. Generating Functions from Recurrences
  14. Cassini's Identity
  15. Fibonacci Idendtities with Matrices
  16. GCD of Fibonacci Numbers
  17. Binet's Formula with Cosines
  18. Lame's Theorem - First Application of Fibonacci Numbers

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