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Why Euler Created Trigonometric Functions

Learn about how Euler introduced trigonometric functions in 1739, revolutionizing calculus and analysis. Discover the significance of sine, transcendental functions, and the differential calculus of logarithmic and exponential quantities. Explore the historical context and impact of Euler's contributions to mathematics.

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Why Euler Created Trigonometric Functions

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  1. Why Euler Created Trigonometric Functions V. Frederick Rickey USMA, West Point NJ-MAA March 31, 2007

  2. What is a sine ? • The Greeks used chords • The Arabs used half-chords • NB: These are line segments, not numbers!

  3. Calculus Differentialis1727 • The Calculus of Finite Differences • The differential Calculus in General • Differentiation of Algebraic Functions • Differentiation of Logarithmic and Exponential Quantities

  4. Draft on Differential Calculus, 1827 • Euler defines functions and then divides them into two classes: • Algebraic • Transcendental • The only transcendental functions are logarithms and exponentials • Euler gives a differential calculus of these functions • NB: no trigonometry

  5. Daniel Bernoulli to Euler, May 4, 1735 • The DE arises in a problem about vibrations on an elastic band. • “This matter is very slippery.”

  6. Euler to Johann BernoulliSeptember 15, 1739 after treating this problem in many ways, I happened on my solution entirely unexpectedly; before that I had no suspicion that the solution of algebraic equations had so much importance in this matter.

  7. Euler creates trig functions in 1739

  8. Linear Differential Equations with constant coefficients • De Integratione Aequationum Differentialium altiorum graduum • 1743 • E62

  9. Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From the preface of the Introductio

  10. Chapter 1: Functions A change of Ontology: Study functions not curves

  11. VIII. Trig Functions

  12. He showed a new algorithm which he found for circular quantities, for which its introduction provided for an entire revolution in the science of calculations, and after having found the utility in the calculus of sine, for which he is truly the author . . . Eulogy by Nicolas Fuss, 1783

  13. Sinus totus = 1 • π is “clearly” irrational • Value of π from de Lagny • Note error in 113th decimal place • “scribam π” • W. W. Rouse Ball discovered (1894) the use of π in Wm Jones 1706. • Arcs not angles • Notation: sin. A. z

  14. Editor’s introduction in 1754 there occurs in analysis a very important type of transcendental quantity, namely the sine . . . which demands a special calculus, which the celebrated author of this dissertation is able rightly to claim all for himself.

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