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Rational Approximations of Sine and Cosine.

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Title: Rational Approximations of Sine and Cosine.
Authors: Azim, Mashrur1 mashrurazimaornab83@gmail.com, Griffin, Christopher2 griffinch@psu.edu
Source: Mathematics Enthusiast. Oct2025, Vol. 22 Issue 3, p335-342. 7p.
Subject Terms: *Calculus, *Algebra, Trigonometric functions, Sine function, Quartic equations
Abstract: In this paper, we use elementary methods to derive a rational function over the integers to approximate the trigonometric sine function on the interval [0, π/2]. This formula can then be used to derive a quartic polynomial with a root close to π/2, providing an interesting algebraic approximation to this value. A more accurate rational function over the reals is then computed using numerical optimization. This new formula, while more accurate, provides a worse approximation of π/2 in the corresponding quartic equation, showing the trade-offs in local vs. global approximation. This paper is accessible to undergraduates and illustrates a combination of mathematical constructions used in Algebra, Calculus and Numerical Optimization. [ABSTRACT FROM AUTHOR]
Copyright of Mathematics Enthusiast is the property of Prof. Bharath Sriraman and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Education Research Complete
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  Data: Rational Approximations of Sine and Cosine.
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  Data: <searchLink fieldCode="JN" term="%22Mathematics+Enthusiast%22">Mathematics Enthusiast</searchLink>. Oct2025, Vol. 22 Issue 3, p335-342. 7p.
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  Data: In this paper, we use elementary methods to derive a rational function over the integers to approximate the trigonometric sine function on the interval [0, π/2]. This formula can then be used to derive a quartic polynomial with a root close to π/2, providing an interesting algebraic approximation to this value. A more accurate rational function over the reals is then computed using numerical optimization. This new formula, while more accurate, provides a worse approximation of π/2 in the corresponding quartic equation, showing the trade-offs in local vs. global approximation. This paper is accessible to undergraduates and illustrates a combination of mathematical constructions used in Algebra, Calculus and Numerical Optimization. [ABSTRACT FROM AUTHOR]
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  Group: Ab
  Data: <i>Copyright of Mathematics Enthusiast is the property of Prof. Bharath Sriraman and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – SubjectFull: Trigonometric functions
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              Text: Oct2025
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