The Derivatives of the Sine and Cosine Functions: A Mini-Primary Source Project for Calculus 1 Students

This curricular module guides students through a method of calculating the derivative of the sine and cosine functions using differentials. It is based on one primary source: Leonhard Euler's Institutiones calculi differentialis (Foundations of Differential Calculus) [Euler 1755], written in 1748. It was the first calculus book to use functions; indeed, Euler himself had been the first mathematician to regularly use an approach which looks like functions to us today about seven years earlier, in his great "pre-calculus'' book, the Introductio in analysin infinitorum (Introduction to the Analysis of the Infinite) [Euler 1748]. While the use of functions makes the material more accessible to our students today, Euler's approach is different enough from that of modern calculus books that it forces students to think carefully about the material.

Jakob Emanuel Handmann 's portrait of Euler, painted while Euler worked on his differential calculus text. Wikimedia Commons.

The major difference in Euler's approach is the lack of limits in his work. The limit concept would not be formally defined and made a part of mathematics for almost a century; Euler based his calculus (following Leibniz) on the differential dx, which was an infinitely small increment of the variable x. While the logical issues in this approach would force 19th-century mathematicians to abandon it (in favor of limits), Euler saw no such issues. (It is the author's experience that Calculus students are closer to Euler on the epistemological spectrum concerning differentials, and are happy to use them for insight; the author further believes that they provide this insight. Fortunately, this belief will be tested by TRIUMPHS' research component!)

Taylor series approximations to sin(x) with polynomials of degree 1, 3, 5, 7, 9, 11 and 13. By IkamusumeFan - Own work, CC BY-SA 3.0, Wikimedia Commons

Perhaps the most surprising aspect of Euler's approach is his use of Taylor series. In fact, he introduced these in the Introductio, and thought of them as a pre-calculus idea. In this way, Euler presaged the later work of Lagrange [Lagrange 1797], who used power series as the starting point for his theory of calculus, and defined the derivative as the “first derived function” in the power series expansion of any given function.

This project may be the first time that students see these series, but they do not need any of the theory of Taylor series in the project. The approximations of sine and cosine via three-term Taylor series are presented as a fait accompli, and students are given an opportunity to convince themselves that the approximation seems valid, even if they can't explain why. It is hoped that this exposure will make Taylor series slightly more approachable when they encounter them in the future, but this is not a major goal of the project.

The project The Derivatives of the Sine and Cosine Functions (pdf) is ready for student use, and the LaTeX source code is available from the author by request.

At the end of the project is a set of instructor notes. These echo some of the material in this introduction, and also include practical advice for the use of the project in the classroom.

This project is the first in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources appearing in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more.  Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection offers additional mini-PSPs and dozens of more extensive full-length PSPs.

Acknowledgments

The development of the student project The Derivatives of the Sine and Cosine Functions has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

Edwards, Harold M. 2007. Euler's definition of the derivative. Bulletin of the American Mathematical Society 44(4): 575–580.

Euler, Leonhard. 1755. Institutiones Calculi Differentialis. St. Petersburg: Academiae Imperialis Scientiarum Petropolitanae. Also in Opera Omnia, Series 1, 10:1–676. English translation by John D. Blanton in Foundations of Differential Calculus, New York: Springer, 2000.

Euler, Leonhard. 1748. Introductio in Analysin Infinitorum. Lausanne: Marcum-Michaelem Bousquet. Also in Opera Omnia, Series 1, 8:1–392. English translation by John D. Blanton in Introduction to Analysis of the Infinite, New York: Springer, 1988.

 Lagrange, Joseph-Louis. 1797. Théorie des fonctions analytiques. Paris: Imprimerie de la République.