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Question Number 163798 by kdaramaths last updated on 10/Jan/22

b y a s i m p l e m e t h o d c a l c u l a t e

\sum_{k = 0}^{n} C O S (k x)

Answered by Ar Brandon last updated on 10/Jan/22

Q 162872

Answered by Kamel last updated on 10/Jan/22

b y a s i m p l e m e t h o d c a l c u l a t e

S_{n} = \sum_{k = 0}^{n} c o s (k x), T_{n} = \underset{k = 0}{\sum^{n}} s i n (k x)

(c o s (x) - 1) S_{n} + s i n (x) T_{n} = \underset{k = 0}{\sum^{n}} c o s ((k - 1) x) = c o s (x) - c o s (n x)

- s i n (x) S_{n} + (c o s (x) - 1) T_{n} = \underset{k = 0}{\sum^{n}} s i n ((k - 1) x) = - s i n (x) - s i n (n x)

∴ {\begin{cases} 2 {s i n}^{2} (\frac{x}{2}) S_{n} - 2 s i n (\frac{x}{2}) c o s (\frac{x}{2}) T_{n} = c o s (n x) - c o s (x) \dots (1) \\ 2 s i n (\frac{x}{2}) c o s (\frac{x}{2}) S_{n} + 2 {s i n}^{2} (\frac{x}{2}) T_{n} = (s i n (x) + s i n (n x)) \dots (2) \end{cases}

(1) . s i n (\frac{x}{2}) + (2) c o s (\frac{x}{2}) \Rightarrow S_{n} = \frac{s i n (\frac{x}{2}) c o s (n x) - s i n (\frac{x}{2}) c o s (x) + c o s (\frac{x}{2}) s i n (x) + c o s (\frac{x}{2}) s i n (n x)}{2 s i n (\frac{x}{2})}

∴ S_{n} = \frac{s i n ((n + \frac{1}{2}) x) + s i n (\frac{x}{2})}{2 s i n (\frac{x}{2})} = \frac{s i n (\frac{n + 1}{2} x) c o s (\frac{n x}{2})}{s i n (\frac{x}{2})}

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