Monday, April 29, 2019

trigonometry - Using De Moivre's Theorem to prove cos(3theta)=4cos3(theta)3cos(theta) trig identity


I am stuck on trying to prove a trig identity using De Moivre's theorem.


I have to prove, cos(3θ)=4cos3(θ)3cos(θ)


I am not sure where to even start, I broke the LHS down to cos(3θ)+isin(3θ)


but I have no idea where to go from here, or if this is fully correct.



If I could get some pointers or a simple worked example that I could follow it would be great.


Thanks


Answer



De Moivre's formula reads (cosθ+isinθ)n=cos(nθ)+isin(nθ) Of course this identity implies the real part should be also equality. That is cos(nθ)={(cosθ+isinθ)n} Hence we have cos(3θ)={cos3θ+3icos2θsinθ3cosθsin2θisin3θ}=cos3θ3cosθsin2θ


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f:AB and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...