## Fibonometry

The term **fibonometry** was coined by John Conway and Alex Ryba in their paper titled, you guessed it, “Fibonometry”. The term describes a freaky parallel between trigonometric formulas and formulas with Fibonacci (F_{n}) and Lucas (L_{n}) numbers. For example, the formula sin(2a) = 2sin(a)cos(a) is very similar to the formula F_{2n} = F_{n}L_{n}. The rule is simple: replace angles with indices, replace sin with F (Fibonacci) and cosine with L (Lucas), and adjust coefficients according to some other rule, which is not too complicated, but I am too lazy to reproduce it. For example, the Pythegorian identity sin^{2}a + cos^{2}a = 1 corresponds to the famous identity L_{n}^{2} – 5F_{n}^{2} = 4(-1)^{n}.

My last year’s PRIMES STEP senior group, students in grades 7 to 9, decided to generalize fibonometry to general Lucas sequences for their research. When the paper was almost ready, we discovered that this generalization is known. Our paper was well-written, and we decided to rearrange it as an expository paper, Fibonometry and Beyond. We posted it at the arXiv and submitted it to a journal. I hope the journal likes it too.

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## Leo B.:

sin(2a) = 2sin(a)cos(b)

b -> a

19 September 2024, 8:43 pm## tanyakh:

Thanks, Leo, I fixed it.

25 September 2024, 5:14 pm