This discussion focuses on deriving trigonometric identities using DeMoivre's Theorem, specifically the identities cos(2θ) = cos²(θ) - sin²(θ) and sin(2θ) = 2sin(θ)cos(θ). Participants explored the expansion of (cos(θ) + i sin(θ))² and equated real and imaginary parts to arrive at the identities. The conversation highlighted common pitfalls in the derivation process and emphasized the importance of careful multiplication and simplification.
PREREQUISITESStudents of mathematics, educators teaching trigonometry, and anyone interested in the applications of complex numbers in deriving trigonometric identities.
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?