# How can we calculate the sum of sines or cosines where the angles are in geometric progression?

For example:

$$\cos\frac{\pi}{7} + \cos\frac{3\pi}{7} + \cos\frac{9\pi}{7}$$

In this example, there are only a few terms, and we can use things like $\cos(9\pi/7) = -\cos(2\pi/7)$ and complex numbers to solve it. I am trying to find if there is a closed formula to a generic case

$$\cos(x) + \cos(x\cdot q) + \cos (x\cdot q^2) + \cdots + \cos(x\cdot q^n)$$

• $$\dfrac{9\pi}7=2\pi—?$$ Use math.stackexchange.com/questions/117114/… – lab bhattacharjee May 1 '18 at 15:48
• I believe there is nothing like a closed formula for the general case; there's no 'simplifying formula' for sums of the form $t^{xq^n}$, after all, and $\cos(xq^n)$ is essentially just a 'special case' of this where $t=e^i$. – Steven Stadnicki May 1 '18 at 16:00