cos 36°

The purpose of this page is to establish that

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.

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,

Triangles ABE and AEP are also similar, in particular

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

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

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

• What Is Trigonometry?
• Sine of a Sum Formula
• Addition and Subtraction Formulas for Sine and Cosine
• Addition and Subtraction Formulas for Sine and Cosine II
• Addition and Subtraction Formulas for Sine and Cosine III
• The Laws of Cosines
• The Law of Sines and Cosines
• Cosine of 36 degrees
• Sine, Cosine, and Ptolemy's Theorem
• arctan(1) + arctan(2) + arctan(3) = π
• arctan(1/2) + arctan(1/3) = arctan(1)
• Morley's Miracle
• Napoleon's Theorem
• A Trigonometric Solution to a Difficult Sangaku Problem

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
   • Getting a Scout out of Desert
9. Fibonacci's Quickies

Golden Ratio

1. Golden Ratio in Geometry
2. Golden Ratio in an Irregular Pentagon
3. Golden Ratio in a Irregular Pentagon II
4. Inflection Points of Fourth Degree Polynomials
5. Wythoff's Nim
6. Inscribing a regular pentagon in a circle - and proving it
7. Cosine of 36 degrees
8. Continued Fractions