# Sums of cos(nx) and sin(nx)

• Nov 15th 2010, 09:11 AM
uberbandgeek6
Sums of cos(nx) and sin(nx)
Find simple formulas for: 1 + cos(x) + cos(2x) + ... + cos(nx)
and: sin(x) + sin(2x) + ... + sin(nx).
Remember that: 1 + a + a^2 + ... + a^n = (a^(n+1) - 1) / (a - 1)
and: (cosx + i*sinx)^n = cos(nx) + i*sin(nx).

So far, I showed that cos(nx) = (cosx + i*sinx)^n - i*sin(nx) and plugged that in for the first sum to get:
1 + cos(x) + cos(2x) + ... + cos(nx)
= 1 + (cosx + i*sinx) - i*sin(x) + (cosx + i*sinx)^2 - i*sin(2x) + ... + (cosx + i*sinx)^n - i*sin(nx)
= ((cosx + i*sinx)^(n+1) - 1)/(cosx + i*sinx - 1) - (i*sin(x) + i*sin(2x) + ... + i*sin(nx))

I'm not sure where to proceed from here to simplify it any further.
• Nov 15th 2010, 09:27 AM
chisigma
The sum of this 'geometric series'...

$\displaystyle \sum_{k=0}^{n} e^{i k x}= \frac{1-e^{i n x}}{1-e^{i x}}$ (1)

... allows You to write...

$\displaystyle \sum_{k=0}^{n} \cos k x= \Re \{\frac{1-e^{i n x}}{1-e^{i x}}\}$ (2)

$\displaystyle \sum_{k=0}^{n} \sin k x = \Im \{\frac{1-e^{i n x}}{1-e^{i x}}\}$ (3)

Kind regards

$\chi$ $\sigma$