The discussion revolves around deriving trigonometric identities from DeMoivre's theorem, specifically focusing on the identities for cos(2θ) and sin(2θ). Participants are exploring the connections between complex exponentials and trigonometric functions.
There is an ongoing exploration of the derivation process, with some participants providing guidance on how to manipulate the expressions. Multiple interpretations of the steps are being considered, and while some clarity is emerging, no consensus has been reached on the complete derivation.
Participants are working under the constraints of deriving identities from DeMoivre's theorem, and there are indications of confusion regarding the manipulation of complex numbers and their corresponding trigonometric forms.
Dick said:Sure. e^(i*2*theta)=(e^(i*theta))^2. Write down both sides and equate the imaginary parts.
Dick said:Let's just call theta t, ok? On one side you have cos(2t)+i*sin(2t). On the other side you have (cos(t)+i*sin(t))*(cos(t)+i*sin(t)). Multiply that out carefully. You are getting things all mushed up.
kathrynag said:cos(2t)+isin(2t)=(cost+isint)^2
cos(2t)+isin(2t)=(cost)^2+2isint-(sint)^2
isin2t=(cost)^2+2isint-(sint)^2-cos(2t)
Is there something I'm missing on what to do here?
kathrynag said:cos(2t)+isin(2t)=(cost+isint)^2
cos(2t)+isin(2t)=(cost)^2+2isintcost-(sint)^2
sin(2t)=2sintcost
Dick said:Yes. And cos(2t)=cos^2(t)-sin^2(t). Right?