Proposition: Inverse Hyperbolic Cosine

The hyperbolic cosine $\mathbb R\to\mathbb R,~x\to\sinh(x)$ is invertible for all positive real numbers $x\ge 0.$ Its inverse function is called the inverse hyperbolic cosine $$\operatorname{arcosh}:[0,\infty[\to[1,\infty[,$$ and can be calculated using the formula $$\operatorname{arcosh}(x)=\ln\left(x+\sqrt{x^2-1}\right).$$

The following graph visualizes the inverse hyperbolic cosine function:

Proofs: 1

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