Trigonometric functions, such as sine and cosine, have historical roots in mathematics, with various methods developed for their calculation. Users inquire about alternative ways to compute these functions without a calculator, leading to discussions on power series, specifically Taylor and Maclaurin series, which can approximate sine and cosine values. The conversation emphasizes that while exact calculations may not be feasible without a calculator, series expansions can provide accurate results to a desired degree of precision. References to historical contexts and mathematical resources are shared to enhance understanding. Overall, the discussion highlights the interplay between historical development and modern computational techniques in trigonometry.