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*}](https://nntdm.net/wp-content/ql-cache/quicklatex.com-2cfc2e8bb6884536753c45ead504214c_l3.png "Rendered by QuickLaTeX.com")
![\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*}](https://nntdm.net/wp-content/ql-cache/quicklatex.com-ea0599823419e3e10d1b33e4770f908a_l3.png "Rendered by QuickLaTeX.com")
Keywords
- Ramanujan type identity
- Trigonometric function
- Cubic equation
- Radicals
2020 Mathematics Subject Classification
- Primary: 11L03
- Secondary: 33B10
References
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- Shiue, P. J.-S., Shannon, A. G., Huang, S. C., & Reyes, J. E. (2022). Notes on efficient computation of Ramanujan cubic equations. Notes on Number Theory and Discrete Mathematics, 28(2), 350–375.
- Shiue, P. J.-S., Shannon, A. G., Huang, S. C., & Reyes, J. E. (2023). A generalized computation procedure for Ramanujan-type identities and cubic Shevelev sum. Notes on Number Theory and Discrete Mathematics, 29(1), 98-129
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.
