Difference between revisions of "Relationship between sin and sinh"

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==Theorem==
<strong>[[Relationship between sin and sinh|Theorem]]:</strong> The following formula holds:
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The following formula holds:
 
$$\sin(z)=-i \sinh(iz),$$
 
$$\sin(z)=-i \sinh(iz),$$
 
where $\sin$ denotes the [[sine]] and $\sinh$ denotes the [[sinh|hyperbolic sine]].
 
where $\sin$ denotes the [[sine]] and $\sinh$ denotes the [[sinh|hyperbolic sine]].
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<strong>Proof:</strong> █
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==Proof==
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From the definition of $\sin$ and $\sinh$ and the [[reciprocal of i]],
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$$-i\sinh(iz) = \dfrac{e^{iz}-e^{-iz}}{2i} =\sin(z),$$
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as was to be shown.
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==References==
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[[Category:Theorem]]
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[[Category:Proven]]

Latest revision as of 05:17, 8 December 2016

Theorem

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

Proof

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

References