- #1
StephenD420
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I need to show that sin i*theta= i* sinh(theta).
where sinh(theta) = .5[e^theta - e^(-theta)]
and cos(theta) = .5[e^theta + e^(-theta)]
and e^(i*theta) = cos(theta) + isin(theta)
if I start with the formula sinh(theta) = .5[e^theta - e^(-theta)]
and plug in e^(i*theta) = cos(theta) + isin(theta)
I get
sinh(theta) = .5*[{cos(theta) + isin(theta)}+e^i - {cos(-theta) + isin(-theta)} + e^i]
since cos(-theta) = cos(theta) and sin(-theta) = -sin(theta)
sinh(theta) = .5*2*i*sin(theta)
or
sinh(theta) = i*sin(theta)
now how do I go from here to
sin(i*theta) = i*sinh(theta)
I know I am almost there I just need a little last nudge.
Thanks
Stephen
where sinh(theta) = .5[e^theta - e^(-theta)]
and cos(theta) = .5[e^theta + e^(-theta)]
and e^(i*theta) = cos(theta) + isin(theta)
if I start with the formula sinh(theta) = .5[e^theta - e^(-theta)]
and plug in e^(i*theta) = cos(theta) + isin(theta)
I get
sinh(theta) = .5*[{cos(theta) + isin(theta)}+e^i - {cos(-theta) + isin(-theta)} + e^i]
since cos(-theta) = cos(theta) and sin(-theta) = -sin(theta)
sinh(theta) = .5*2*i*sin(theta)
or
sinh(theta) = i*sin(theta)
now how do I go from here to
sin(i*theta) = i*sinh(theta)
I know I am almost there I just need a little last nudge.
Thanks
Stephen
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