Thread: complex variables

How do I express sin(e^i) in terms of z = x + yi with x,y in R.

I say that e^i = cos1 + i*sin1 (since e^z = e^x(cosy + i*siny).
Then sin(cos1 + i*sin1) = [e^(i*(cos1 + i*sin1)) - e^(-i*(cos1 + i*sin1))]/2i, which equals (e^(i*cos1 - sin1) - e^(-i*cos1 + sin1))/2i. How do I write this in the form x + yi?

Thanks for your help in advance.

Originally Posted by PvtBillPilgrim

How do I express sin(e^i) in terms of z = x + yi with x,y in R.

I say that e^i = cos1 + i*sin1 (since e^z = e^x(cosy + i*siny).
Then sin(cos1 + i*sin1) = [e^(i*(cos1 + i*sin1)) - e^(-i*(cos1 + i*sin1))]/2i, which equals (e^(i*cos1 - sin1) - e^(-i*cos1 + sin1))/2i. How do I write this in the form x + yi?

Thanks for your help in advance.

note that if a complex number is given in the form then we can find its rectangular form using the following relationships:

and

Originally Posted by PvtBillPilgrim

How do I express sin(e^i) in terms of z = x + yi with x,y in R.

Here are two that may help you.