Difference between revisions of "Relationship between sinh and sin"

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(Proof)
 
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==Proof==
 
==Proof==
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By definition,
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$$\sinh(z) = \dfrac{e^{z}-e^{-z}}{2},$$
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and so by the definition of $\sin$ and the [[reciprocal of i]], we see
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$$-i\sinh(iz)=\dfrac{e^{iz}-e^{-iz}}{2i},$$
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as was to be shown. █
  
 
==References==
 
==References==
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Coth|next=Relationship between cosh and cos}}: $4.5.7$
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
[[Category:Unproven]]
+
[[Category:Proven]]

Latest revision as of 03:49, 8 December 2016

Theorem

The following formula holds: $$\sinh(z)=-i\sin(iz),$$ where $\sinh$ is the hyperbolic sine and $\sin$ is the sine.

Proof

By definition, $$\sinh(z) = \dfrac{e^{z}-e^{-z}}{2},$$ and so by the definition of $\sin$ and the reciprocal of i, we see $$-i\sinh(iz)=\dfrac{e^{iz}-e^{-iz}}{2i},$$ as was to be shown. █

References