Is the product of two sine functions always real-valued?

[Edwin]

Hi, I was playing around with Euler's Identity, and I found something (or at least I think I found something) interesting:

It is a well known identity
sin(z) = [exp(iz) - exp(-iz)]/(2*i), where z is any complex number, exp is the complex exponential function, and i is the imaginary constant.

So if we bring the 2*i upstairs, we get

2*i*sin(z) = exp(iz) - exp(-iz) (2)

From (2), we can see that

-4*sin(z)*sin(w) = [exp(iz) - exp(-iz)]*[exp(iw) - exp(-iw) ]

= exp[i(z+w)] +exp[-i(z+w)] - (exp[i(z-w)] + exp[-i(z-w)] )
Since exp[-i(z+w)] is the complex conjugate of exp[i(z+w)], and exp[-i(z-w)] is the complex conjugate of exp[i(z-w)], we have


exp[i(z+w)] +exp[-i(z+w)] - (exp[i(z-w)] + exp[-i(z-w)] )

=2RealPart{exp[i(z+w)]} - 2RealPart{exp[i(z-w)]}, which is real for every complex w and z.

Is this really true? Am I making some mistake here?

Inquisitively,

Edwin G. Schasteen

 

[Hurkyl]

e^ix and e^-ix are complex conjugates when x is real, but not often otherwise.

 

[Edwin]

Hi Hurkyl,

Thank you! That makes a lot more sense now.

Best Regards,

Edwin