Difference between revisions of "Relationship between tanh, inverse Gudermannian, and sin"
From specialfunctionswiki
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The following formula holds: | The following formula holds: | ||
$$\mathrm{tanh}(\mathrm{gd}^{-1}(x))=\sin(x),$$ | $$\mathrm{tanh}(\mathrm{gd}^{-1}(x))=\sin(x),$$ | ||
| − | where $\mathrm{tanh}$ is the [[tanh|hyperbolic tangent]], $\mathrm{gd}^{-1}$ is the [[inverse Gudermannian]], and $\sin$ is | + | where $\mathrm{tanh}$ is the [[tanh|hyperbolic tangent]], $\mathrm{gd}^{-1}$ is the [[inverse Gudermannian]], and $\sin$ is [[sine]]. |
==Proof== | ==Proof== | ||
Revision as of 02:31, 21 December 2016
Theorem
The following formula holds: $$\mathrm{tanh}(\mathrm{gd}^{-1}(x))=\sin(x),$$ where $\mathrm{tanh}$ is the hyperbolic tangent, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\sin$ is sine.
