Hyperbolic versions of latest posts

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The post A curious trig identity contained the theorem that for real x and y,

|sin(x + iy)| = |sin x + sin iy|

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.

renewcommand{arraystretch}{2.2} begin{array}{c|c|c|c} & sinh^{-1} & cosh^{-1} & tanh^{-1} \ hline sinh & x & sqrt{x^{2}-1} & dfrac{x}{sqrt{1-x^2}} \ hline cosh & sqrt{x^{2} + 1} & x & dfrac{1}{sqrt{1 - x^2}} \ hline tanh & dfrac{x}{sqrt{x^{2}+1}} & dfrac{sqrt{x^{2}-1}}{x} & x \ end{array}

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

The post Hyperbolic versions of latest posts first appeared on John D. Cook.

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