Functions of Complex variables

Question

Sin(Θ+iΦ)=cosα+i*sinα

prove that 

cos^4(Θ)=sin^2(α)=sinh^4(Φ)

Answer 1

Hi,

This is similar to your previous question

sin(Θ+iΦ)=cosα+i*sinα

 

sin(Θ)cosh(Φ)+i cos(Θ)sinh(Φ)=cosα+i*sinα

 

Calculate Modul from both sides

sin²(Θ)cosh²(Φ)+ cos²(Θ)sinh²(Φ)=1

From here you can find cos²(Θ):

(cos²(Θ)+sin²(Θ))cosh²(Φ)=1- cos²(Θ)

 

 

 cos²(Θ)=1-cosh²(Φ)=-sinh²(Φ)

So,

cos⁴(Θ)=sinh⁴(Φ)

 

There are different ways to prove. Another way: You can first square both sides and the separate Re and Im parts.

Again, I did not give a full answer but I hope it would help you. I'm just trying to give you direction. 

But any questions, feel free to write me.