Given-f-x-k-0-n-n-C-k-sin-kx-cos-n-k-x-Find-a-simple-form-for-f-x-Your-answer-should-be-written-like-c-n-g-nx-

Question Number 50908 by Smail last updated on 22/Dec/18

G i v e n f (x) = \underset{k = 0}{\sum^{n}}^{n} C_{k} s i n (k x) c o s ((n - k) x)

F i n d a s i m p l e f o r m f o r f (x)

(Y o u r a n s w e r s h o u l d b e w r i t t e n l i k e c (n) . g (n x))

Answered by Smail last updated on 23/Dec/18

f (x) = \underset{k = 0}{\sum^{n}}^{n} C_{k} s i n (k x) c o s ((n - k) x)

l e t l = n - k

f (x) = \underset{l = 0}{\sum^{n}}^{n} C_{n - l} s i n ((n - l) x) c o s (l x)

= \underset{l = 0}{\sum^{n}}^{n} C_{l} s i n ((n - l) x) c o s (l x)

2 f (x) = \underset{k = 0}{\sum^{n}}^{n} C_{k} (s i n ((n - k) x) c o s (k x) + s i n (k x) c o s ((n - k) x))

= \underset{k = 0}{\sum^{n}}^{n} C_{k} (s i n ((n - k) x + k x))

2 f (x) = \underset{k = 0}{\sum^{n}}^{n} C_{k} s i n (n x) = 2^{n} s i n (n x)

f (x) = 2^{n - 1} s i n (n x)

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