Две тригонометрични суми

[KOPMOPAH]

Да се намерят сумите:$$S=\sum_{k=0}^{n}\sin(y+kx) \\ C=\sum_{k=0}^{n}\cos(y+kx)$$

[S.B.]

Да се намерят сумите:$$S=\sum_{k=0}^{n}\sin(y+kx) \\ C=\sum_{k=0}^{n}\cos(y+kx)$$

[tex]S = \sum_{k=0}^{n }sin(y + kx) = siny + sin(y + x) + sin(y + 2x) + sin(y + 3x) + ..........sin(y + nx) =[/tex]
[tex]= siny + siny.cosx + cosy.sinx + siny.cos2x + cosy.sin2x + siny.cos3x + sin3x.cosy + ........... + siny.cosnx + sinnx.cosy =[/tex]
[tex]= siny(1 + cosx + cos2x + cos3x....+ cosnx) + cosy(sinx + sin2x + sin3x + ......+ sinnx ) =[/tex]
[tex]= siny(1 + \displaystyle\frac{sin\displaystyle\frac{(n + 1)x}{2}.cos\displaystyle\frac{nx}{2}}{sin\displaystyle\frac{x}{2}}) + cosy(\displaystyle\frac{sin\displaystyle\frac{(n + 1)x}{2}.sin\displaystyle\frac{nx}{2}}{sin\displaystyle\frac{x}{2}})[/tex]

[tex]C = \sum_{k=0}^{n }cos(y + kx) = cosy + cos(y + x) + cos(y + 2x ) + cos(y + 3x) + ........ + cos(y + nx) =[/tex]
[tex]= cosy + cosy.cosx - siny.sinx + cosy.cos2x - siny.sin2x + cosy.cos3x - siny.sin3x +.........+ cosy.cosnx - siny.sinnx =[/tex]
[tex]= cosy(1 + cosx + cos2x + cos3x + ......+ cosnx) - siny(sinx + sin2x + sin3x + .......sinnx) =[/tex]
[tex]= cosy( 1 + \displaystyle\frac{sin\displaystyle\frac{(n + 1)x}{2}.cos\displaystyle\frac{nx}{2}}{sin\displaystyle\frac{x}{2}}) - siny(\displaystyle\frac{sin\displaystyle\frac{(n + 1)x}{2}.sin\displaystyle\frac{nx}{2}}{sin\displaystyle\frac{x}{2}})[/tex]

[Sup3rlum]

Да се намерят сумите:$$S=\sum_{k=0}^{n}\sin(y+kx) \\ C=\sum_{k=0}^{n}\cos(y+kx)$$

Разглеждаме $E=\sum_{k=0}^ne^{i(y+kx)}=C+iS$

$E=e^{iy}\sum_{k=0}^ne^{ikx}=e^{iy}\frac{e^{ix(n+1)}-1}{e^{ix}-1}=\frac{e^{i(x(n+1)+y)}-e^{iy}}{e^{ix}-1}=-ie^{-i\frac{x}{2}}\frac{e^{i(x(n+1)+y)}-e^{iy}}{2sin(\frac{x}{2})}=\frac{-ie^{i(x(n+1)+y-\frac{x}{2})}+ie^{i(y-\frac{x}{2})}}{2sin(\frac{x}{2})}=\frac{-i(cos(x(n+1)+y-\frac{x}{2})+isin(x(n+1)+y-\frac{x}{2}))+i(cos(y-\frac{x}{2})+isin(y-\frac{x}{2}))}{2sin(\frac{x}{2})}$

$\Rightarrow C=Re(E)=\frac{sin(x(n+\frac{1}{2})+y)-sin(y-\frac{x}{2})}{2sin(\frac{x}{2})}$
$\Rightarrow S=Im(E)=\frac{-cos(x(n+\frac{1}{2})+y)-cos(y-\frac{x}{2})}{2sin(\frac{x}{2})}$