Circular and hyperbolic functions differ by rotations

This article explores the mathematical relationship between circular (trigonometric) and hyperbolic functions, demonstrating that they differ by a rotation in the complex plane. It provides key identities such as cosh(z)=cos(iz) and sinh(z)=-i sin(iz), and explains how each trig function can be transformed into its hyperbolic counterpart by applying a function to iz and multiplying by a constant. The post also categorizes equations based on whether the left side involves foo(iz), fooh(iz), foo(z), or fooh(z). Written as a warm-up for a follow-up post on couth and uncouth function pairs, this content is relevant to mathematics and computer science, particularly for those interested in complex analysis and function transformations.