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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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