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Question

Mathematics Question on Trigonometric Functions

sin (ix)=sin\, (ix) =

A

i sin hxi\, sin\, hx

B

i sin xi\, sin\, x

C

sin xsin\, x

D

sin hxsin\, hx

Answer

i sin hxi\, sin\, hx

Explanation

Solution

sin(ix)=ei(ix)−e−i(ix)2isin \left(ix\right)=\frac{e^{i\left(ix\right)}-e^{-i\left(ix\right)}}{2i}
[sin x=eix−e−ix2i]\left[sin \,x=\frac{e^{ix}-e^{-ix}}{2i}\right]
=e−x−ex2i=−i2(e−x−ex)=\frac{e^{-x}-e^{x}}{2i}=-\frac{i}{2}\left(e^{-x}-e^{x}\right)
=i(ex−e−x2)=i\left(\frac{e^{x}-e^{-x}}{2}\right)
=i sin hx=i\,sin \,hx