Monday, August 14, 2017

trigonometry - Finding a closed form for $cos{x}+cos{3x}+cos{5x}+cdots+cos{(2n-1)x}$




We have to find





$$g(x)=\cos{x}+\cos{3x}+\cos{5x}+\cdots+\cos{(2n-1)x}$$




I could not get any good idea .



Intialy I thought of using



$$\cos a+\cos b=2\cos(a+b)/2\cos (a-b)/2$$


Answer




Let $z=\cos\theta+i\sin\theta$ i.e. $z=e^{i\theta}$



Your sum:$$e^{i\theta}+e^{3i\theta}+e^{5i\theta}+...e^{(2n-1)i\theta}$$



This is a GP with common ratio $e^{2i\theta}$



Therefore sum is $$\frac{a(r^n-1)}{r-1}$$
$$\frac{e^{i\theta}(e^{2ni\theta}-1)}{e^{2i\theta}-1}$$
$$\frac{(\cos \theta+i\sin\theta)(\cos(2n\theta)+i\sin\theta-1)}{\cos(2\theta)+i\sin(2\theta)-1}$$




Computing it's real part should give you the answer



Acknowledgement:Due credits to @LordShark Idea


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection $f \colon A \rightarrow B$ and I want to get bijection. Can I just resting codomain to $f(A)$? I know that every function i...