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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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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 $\,\exp(z).$

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