I have just started learning about trigonometric ratios of complex arguments but I couldn't find any justification or derivation for extending trigonometric ratios to complex field. Also the Euler's relationship is for $exp(x+iy)$ where x and y are real so putting $exp(iz)$ where z is complex like that and defining ratios in complex arguments , i wasn't able to digest. Can anyone please tell how,when and why were trigonometric ratios of complex arguments defined and their development in course of time. There would be some reason to do this and also to not change the names because these names where historically defined for triangles. I think the intuition for its historical development may help me in grasping the concept.
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1$\begingroup$ Googling {Euler trigonometry "complex numbers" "history"}, I found within the first few hits the following, which should be useful: [1] A Short History of Complex Numbers by Orlando Merino (January 2006). (continued) $\endgroup$– Dave L RenfroJan 9, 2021 at 18:36
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1$\begingroup$ [2] Complex numbers and Trigonometric Identities (author not identified, nor does URL-shortening help). [3] De Moivre's Theorem by Cynthia Schneider (Masters thesis, Fall 2011). $\endgroup$– Dave L RenfroJan 9, 2021 at 18:36
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$\begingroup$ @ Dave L Renfro the given PDFs don't address the issue in question . The first one just tells about when complex numbers where discovered while the other 2 are much more like some college study material with formula . They don't tell how and why complex trigonometric functions came. $\endgroup$– Guji2203Jan 10, 2021 at 6:37
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1$\begingroup$ I think once you have Euler's formula $e^{i\theta} = \cos \theta + i\sin \theta,$ it's straightforward to solve for $\cos \theta$ and $\sin \theta$ as a rational function of $e^{i\theta}$ and $e^{-i\theta}$ and then use these expressions to obtain trig. values of non-real values of $\theta.$ Surely Euler did this somewhere (why I included "Euler" in the google search), and certainly Gauss and Cauchy made use of complex-inputs to trig. functions. $\endgroup$– Dave L RenfroJan 10, 2021 at 8:12
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$\begingroup$ Of possible interest is Alexander John Ellis (1814-1890)'s 1843 translation (especially in the vicinity of p. 67), titled The Spirit of Mathematical Analysis, and Its Relation to a Logical System, of Martin Ohm (1792-1872)'s 1842 book Der Geist der Mathematischen Analysis und ihr Verhältniss zur Schule. $\endgroup$– Dave L RenfroJan 10, 2021 at 8:14
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I am reasonably certain that the answer to your question about complex trigonometric functions such as $\,\sin(z)\,$ goes back to the late 1800s when complex analysis was brought to an advanced state. The idea is that some well behaved real functions can be extended uniquely using the analytic continuation process to the whole complex plane (with some possible poles allowed). These functions include polynomial and rational functions, circular and hyperbolic trigonometric functions, and many special functions including the gamma function. One of the simplest elementary functions is the exponential function $\,\exp(z)\,$ which has an imaginary period of $\,2\pi i.\,$ The circular or hyperbolic trigonometric functions are then defined as certain rational functions of $\,\exp(iz)\,$ or $\,\exp(z).$
