General Sine and Cosine formula for sum of a finite number of angles

Question

I was wondering is there is a general formula for $\sin(x_1+x_2+x_3+...+x_n)$ as well as for the cosine function. I know that $\sin(x_1+x_2)=\sin(x_1)\cos(x_2)+\cos(x_1)\sin(x_2)$ and $\cos(x_1+x_2)=\cos(x_1)\cos(x_2)-\sin(x_1)\sin(x_2)$ But I want to find a general formula for the sum of a finite number of angles for the Sine and the cosine but I didn't noticed any pattern. I suspect that it may have a recursive pattern. Any suggestions and hints (not answers) will be appreciated.

Answer 1

Well, for $n=3$: $$\cos(x+y+z)=\cos(x+y)\cos(z)-\sin(x+y)\sin(z)=(\cos(x)\cos(y)-\sin(x)\sin(y))\cos(z)-(\sin(x)\cos(y)+\cos(x)\sin(y))\sin(z).$$ It should be easy to do this for higher $n$ as well, and for $\sin(x+y+z)$. It resembles some sort of cyclic sum, though I'm not quite sure how.

Answer 2

You could use a pair of two mutually recursive formulas for both $\sin(\sum x_i)$ and $\cos(\sum x_i)$. Both formulas can be derived from simple sine/cosine of sum of two elements:

$\sin(x_1+...+x_N + x_{N+1}) = \sin(x_1 + ... + x_N) \cdot \cos x_{N+1} + \cos(x_1 + ... + x_N) \cdot \sin x_{N+1}$

$\cos(x_1+...+x_N + x_{N+1}) = \cos(x_1 + ... + x_N) \cdot \cos x_{N+1} - \sin(x_1 + ... + x_N) \cdot \sin x_{N+1}$