Difference between revisions of "Relationship between sin and sinh"
From specialfunctionswiki
| Line 5: | Line 5: | ||
==Proof== | ==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== | ==References== | ||
[[Category:Theorem]] | [[Category:Theorem]] | ||
| − | [[Category: | + | [[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.
