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I know that sine and cosine functions are used ubiquitously today, but I got to wondering about what sort of unsolved or difficult to solve problems required inventing the trigonometric functions.

For instance, the complex number $i$ was invented for the purpose of solving quadratics unsolvable in the reals - i.e. solving the equation $x^2=-1$. The number $0$ was invented to represent absence, making possible much more efficient, unambiguous, and reliable number keeping. The Laplace Transform was invented to transform entire difference equations and make solving DEs much easier and faster in many cases.

What problem did someone have to invent the sine and cosine functions to solve, or to make much easier?

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Trigonometric functions were invented by astronomers to solve problems of spherical trigonometry. The main problem was passing from the ecliptic to equatorial coordinate system and back. Plane trigonometry was a secondary aplication which appeared as a byproduct. The original trigonometric function was the chord, $\mathrm{chd}\, t=2\sin(t/2)$. The first table of chords was made by the Greek astronomer Hipparchus. The first table of chords which survived is due to another astronomer, Ptolemy. In the medieval times trigonometry was developed in India, they introduced the modern functions, sine and cosine. It is not known who exactly and when introduced them. (See Wikipedia, "Surya Siddhanta"). Later the Arabs, or Indians added tangent. All this development happened for the needs of astronomy.

The statement that $i=\sqrt{1}$ was invented to solve quadratic equations which have no real solutons is not quite correct. People were happy with some quadratic equations having solutions, and others not. $i$ was invented to solve cubic equations. Cardan's formula involves square roots and cubic roots. When you try to solve a simple cubic equation with three real roots with Cardan's formula, for example, $x^3-x=x(x-1)(x+1)=0$, the square root in Cardan's formula has to be extracted from a negative quantity. So to obtain the correct result from Cardans's formula in such case, one needs to compute with square roots of negative numbers.

EDIT. Until the late 1970s astronomy and geodesy remained the most important areas of application of trigonometric functions. Every ship (and a trans-ocean airplane) had a navigator whose main job was to determine the position of the ship by astronomical observations. He used trigonometry, to solve essentially the same problem for which trigonometry was invented. (He used tables of logarithms of sines or tables specially designed for this specific problem).

In the late 1970-s two revolutions happened, almost simultaneosly, which eliminated this profession. 1. Electronic calculators (and later computers), and almost at the same time 2. Satellite technology. These made the profession of navigator obsolete. Nowadays there is no mass profession where the working knowledge of trigonometric functions is required.

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    $\begingroup$ Concerning your last sentence, there is such a mass profession: high school math teacher. :) $\endgroup$ – KCd Mar 2 '16 at 12:21
  • $\begingroup$ @KCd: you are right. But this is slowly reduced, faster in some countries, slower in others. $\endgroup$ – Alexandre Eremenko Mar 2 '16 at 14:21

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