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Verify Identities Cos(z) = (exp(iz)+exp(-iz))/2 sin(z)= (exp(iz)-exp(-iz))/2i

Using identities show that sin(z) and cos(z) are surjective from C to C.

Determine all z in C such that sin(z) = 12i/5

Solutions: The verification seems simple using eulers formula and the even/odd nature of cos/sin. My problem is using the identities to show sin/cos are surjective.

Attempt:

Given m in C, m = cos(z) = (exp(iz)+exp(-iz))/2 implying 2m = exp(iz)+exp(-iz) implying Log(2m) = Log(exp(iz)(1+exp(-2iz)) = izLog(1+exp(-2iz). From here I was going to expand the log into a taylor series.. though I'm not sure how that will help. Any suggestions would be appreciated. Obviously, once I can find some equation to determine z, I can just plug it in to find z for 12i/5 and then add steps of 2ipi since Log is periodic over the complex numbers.

Just spent around 2 hours trying to tackle the problem from a different perspective... not sure how to proceed. Any help, if just a possible hint from someone who is unsure themselves would be appreciated.

Using identities show that sin(z) and cos(z) are surjective from C to C.

Determine all z in C such that sin(z) = 12i/5

Solutions: The verification seems simple using eulers formula and the even/odd nature of cos/sin. My problem is using the identities to show sin/cos are surjective.

Attempt:

Given m in C, m = cos(z) = (exp(iz)+exp(-iz))/2 implying 2m = exp(iz)+exp(-iz) implying Log(2m) = Log(exp(iz)(1+exp(-2iz)) = izLog(1+exp(-2iz). From here I was going to expand the log into a taylor series.. though I'm not sure how that will help. Any suggestions would be appreciated. Obviously, once I can find some equation to determine z, I can just plug it in to find z for 12i/5 and then add steps of 2ipi since Log is periodic over the complex numbers.

Just spent around 2 hours trying to tackle the problem from a different perspective... not sure how to proceed. Any help, if just a possible hint from someone who is unsure themselves would be appreciated.

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