# Binet's Formula with Cosines

Elsewhere we found cosines and sines of several angles related to the regular pentagon.

\(\mbox{cos}(36^{\circ}) = \frac{(1 + \sqrt{5})}{4}.\)

For the following I shall need one more:

\(\mbox{cos}(108^{\circ}) = -\mbox{cos}(72^{\circ})=-\mbox{sin}(18^{\circ})=\frac{(1 - \sqrt{5})}{4}.\)

Observe that \(36^{\circ} = \frac{\pi}{5}\) and \(108^{\circ} = \frac{3\pi}{5}\). On the other hand, \(\frac{(1 + \sqrt{5})}{4}=\frac{\phi}{2}\), half of the Golden Ratio, while \(\frac{(1 - \sqrt{5})}{4}=\frac{1}{2}[-\frac{1}{\phi}]\).

What I plan to do is to combine those values to rewrite the Binet's formula for the Fibonacci numbers:

\(F_{n}=\frac{1}{\sqrt{5}}({\phi}^{n}-{\tau}^{n}),\)

where \(\tau\) is exactly \(-\frac{1}{\phi}\).

The combination of the above obtained, according to [Grimaldi, p. 67], first by W. Hope-Jones in 1921, leads to

\(F_{n}=\frac{1}{\sqrt{5}}(2^{n})\left[\mbox{cos}^{n}(\frac{\pi}{5})-\mbox{cos}^{n}(\frac{3\pi}{5})\right].\)

### References

- R. Grimaldi,
*Fibonacci and Catalan Numbers: an Introduction*, Wiley, 2012

### Trigonometry

- What Is Trigonometry?
- Addition and Subtraction Formulas for Sine and Cosine
- The Law of Cosines (Cosine Rule)
- Cosine of 36 degrees
- Tangent of 22.5
^{o}- Proof Wthout Words - Sine and Cosine of 15 Degrees Angle
- Sine, Cosine, and Ptolemy's Theorem
- arctan(1) + arctan(2) + arctan(3) = π
- Trigonometry by Watching
- arctan(1/2) + arctan(1/3) = arctan(1)
- Morley's Miracle
- Napoleon's Theorem
- A Trigonometric Solution to a Difficult Sangaku Problem
- Trigonometric Form of Complex Numbers
- Derivatives of Sine and Cosine
- ΔABC is right iff sin²A + sin²B + sin²C = 2
- Advanced Identities
- Hunting Right Angles
- Point on Bisector in Right Angle
- Trigonometric Identities with Arctangents
- The Concurrency of the Altitudes in a Triangle - Trigonometric Proof
- Butterfly Trigonometry
- Binet's Formula with Cosines
- Another Face and Proof of a Trigonometric Identity
- cos/sin inequality
- On the Intersection of kx and |sin(x)|
- Cevians And Semicircles
- Double and Half Angle Formulas
- A Nice Trig Formula
- Another Golden Ratio in Semicircle
- Leo Giugiuc's Trigonometric Lemma
- Another Property of Points on Incircle
- Much from Little
- The Law of Cosines and the Law of Sines Are Equivalent
- Wonderful Trigonometry In Equilateral Triangle

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