Difference between revisions of "Relationship between tanh, inverse Gudermannian, and sin"
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| − | + | ==Theorem== | |
| − | + | 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 the [[sine]]. | where $\mathrm{tanh}$ is the [[tanh|hyperbolic tangent]], $\mathrm{gd}^{-1}$ is the [[inverse Gudermannian]], and $\sin$ is the [[sine]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
Revision as of 00:41, 4 June 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 the sine.
