The trigonometric and hyperbolic functions share a deep structural affinity rooted in the Euler formula. This note demonstrates that three classical trigonometric inequalities admit direct hyperbolic analogues obtained by the formal substitution x↦ix. Using elementary differential calculus and monotonicity arguments, we provide complete, self-contained proofs of the resulting inequalities. The examples illustrate that analogical reasoning, while not a substitute for rigorous verification, serves as a fruitful heuristic for discovering new hyperbolic counterparts of known trigonometric results.
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Hongchang Li (2026) studied this question.
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