Hyperbolic versions of latest posts
by John from John D. Cook on (#73V97)
The post A curious trig identity contained the theorem that for real xandy,
This theorem also holds when sine is replaced with hyperbolic sine.
The post Trig of inverse trig contained a table summarizing trig functions applied to inverse trig functions. You can make a very similar table for the hyperbolic counterparts.
The following Python code doesn't prove that the entries in the table are correct, but it likely would catch typos.
from math import * def compare(x, y): print(abs(x - y) < 1e-12) for x in [2, 3]: compare(sinh(acosh(x)), sqrt(x**2 - 1)) compare(cosh(asinh(x)), sqrt(x**2 + 1)) compare(tanh(asinh(x)), x/sqrt(x**2 + 1)) compare(tanh(acosh(x)), sqrt(x**2 - 1)/x) for x in [0.1, -0.2]: compare(sinh(atanh(x)), x/sqrt(1 - x**2)) compare(cosh(atanh(x)), 1/sqrt(1 - x**2))
Related post: Rule for converting trig identities into hyperbolic identities
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