Difference between revisions of "Gudermannian"
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Revision as of 22:48, 25 August 2015
The Gudermannian $\mathrm{gd}$ is defined for $x \in \mathbb{R}$ by the formula $$\mathrm{gd}(x) = \displaystyle\int_0^x \dfrac{1}{\cosh t} dt$$
Properties
Theorem
The following formula holds: $$\sin(\mathrm{gd}(x))=\tanh(x),$$ where $\sin$ denotes the sine, $\mathrm{gd}$ denotes the Gudermannian, and $\tanh$ denotes the hyperbolic tangent.
Proof
References
Theorem
The following formula holds: $$\cos(\mathrm{gd}(x))=\mathrm{sech}(x),$$ where $\cos$ denotes the cosine, $\mathrm{gd}$ denotes the Gudermannian, and $\mathrm{sech}$ denotes the hyperbolic secant.
