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 of 360°/5, i.e.

∠CAD = 36°.

By symmetry ∠BAC = ∠DAE implying that this 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. (This calculations were in fact performed to justify a construction of regular pentagon. We return to them here in order to facilitate the references.)

In ΔAEP 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

from cos(2α) = 2cos²(α) - 1 and then

from cos²(α) + sin²(α) = 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

from sin(2α) = 2sin(α)cos(α).

We can easily find cos(72°) from cos(2α) = 2cos²(α) - 1:

cos(72°) = 2cos²(36°) - 1

= 2[(√5 + 1) / 4]² - 1

= (6 + 2√5) / 8 - 1

= (3 + √5) / 4 - 1

= (√5 - 1) / 4.

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

And, of course,

because sin(72°) = cos(18°).

Trigonometry

Fibonacci Numbers

Golden Ratio

40608073