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∘=360∘5, 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 ABP, ABE, AEP 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 △AEP, AE=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=2cos2α−1 and then

sin(18∘)=2(3−5–√)−−−−−−−−√/4

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

sin(18∘)=(5–√−1)/4,

which is immediately verified by squaring the two expressions. Also

sin(36∘)=2(5−5–√)−−−−−−−−√/4

from sin2α=2 sinα cosα.

We can easily find cos(72∘) from cos2α=2cos2α−1:

cos(72∘)

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

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

Golden Ratio in Geometry

Diseño Electrónico

miércoles, 11 de septiembre de 2024

Sumatoria de ángulos en un poligono de n vertices, n angulos = 180°*(n-2)

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

cos 36°

Trigonometry

Fibonacci Numbers