On Ramanujan type identities and Cardano formula

Kai Wang
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 3, Pages 155–174
DOI: 10.7546/nntdm.2021.27.3.155-174
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Kai Wang
2346 Sandstone Cliffs Dr, Henderson NV, USA

Abstract

In this paper we will prove some Ramanujan type identities such as

    \begin{align*} &\sqrt[3]{\sin\left(\frac{\pi}{9}\right)} + \sqrt[3]{\sin\left(\frac{2\pi}{9}\right)} + \sqrt[3]{\sin\left(\frac{14\pi}{9}\right)} \\ & = \left(-\frac{\sqrt[18]{3}}{2}\right) \left(\sqrt[3]{6+3\left(\sqrt[3]{6-3\sqrt[3]{9}}+ \sqrt[3]{3-3\sqrt[3]{9}}\right)}\right), \end{align*}

    \begin{align*} &\sqrt[3]{\tan\left(\frac{\pi}{9}\right)} + \sqrt[3]{\tan\left(\frac{4\pi}{9}\right)} + \sqrt[3]{\tan\left(\frac{7\pi}{9}\right)} \\ & = \left(-\sqrt[18]{3}\right) \left(\sqrt[3]{-3\sqrt[3]{3}+6+3(\sqrt[3]{\!21 - 3(3\sqrt[3]{3}\!-\!\sqrt[3]{9}) } - \sqrt[3]{\!3 + 3(3\sqrt[3]{3}\!+\!\sqrt[3]{9})})}\right). \end{align*}

Keywords

  • Ramanujan type identity
  • Trigonometric function
  • Cubic equation
  • Radicals

2020 Mathematics Subject Classification

  • Primary: 11L03
  • Secondary: 33B10

References

  1. Berndt, B., & Bhargava, S. (1993). Ramanujan–for lowbrows, American Mathematical Monthly, 100, 644–656.
  2. Ramanujan, S. (1957). Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay.
  3. Shevelev, V. (2009). On Ramanujan cubic polynomials. South East Asian Journal of Mathematics and Mathematical Sciences, 8(1), 113–122.
  4. Wikipedia contributors. (2021, February 2). Cubic function. In Wikipedia, The Free Encyclopedia. Retrieved August 2, 2021, from https://en.wikipedia.org/w/index.php?title=Cubic_function&oldid=1004476794.
  5. Wikipedia contributors. (2021, July 30). Heptagonal triangle. In Wikipedia, The Free Encyclopedia. Retrieved August 2, 2021, from https://en.wikipedia.org/w/index.php?title=Heptagonal_triangle&oldid=1036177823.
  6. Wituła, R. (2009). Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2π=7. Journal of Integer Sequences, 12, Article 09.8.5.
  7. Wituła, R. (2012). Ramanujan type trigonometric formulae. Demonstratio Mathematica, XLV(4), 779–796.
  8. Wróbel, A., Hetmaniok, E., Pleszczyński, M., & Wituła, R. (2016). On improvement of the numerical application for Cardano’s formula in Mathematica software, Symposium for Young Scientists in Technology, Engineering and Mathematics, Catania, Italy, September 27–29, 2015, 71–78. Available online at http://ceur-ws.org/Vol-1543/p10.pdf.

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Cite this paper

Wang, K. (2021). On Ramanujan type identities and Cardano formula. Notes on Number Theory and Discrete Mathematics, 27(3), 155-174, DOI: 10.7546/nntdm.2021.27.3.155-174.

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