Summation of trigonometric functions such as $\sin x$ I am currently studying Integration (a very basic introduction) and I have a question regarding the summation of trigonometric functions. Given $f(x) = \sin x$, determine the area under the curve between a and b. By definition of a definite integral (using sigma notation rather than antiderivatives),
$$
\int_a^b f(x)\,dx = \lim_{n\to\infty} \sum_{i=1}^\infty f(x_i)\,\Delta x
$$
Before I proceed in determining the integral, is there a way that I can determine the value of
$$
\sum_{i=1}^n\sin i
$$
To put this in context, I am wondering if there is a formula that one can use for the summation of trig functions. There are formulas for summation of polynomials. i.e
$$
\sum_{i=1}^n i = \frac{n(n+1)}{2}
$$
I would like to expand this method to all trig functions! 
Any help would be much appreciated.

Consider $$S=\sum_{k=1}^n\sin (k)$$ $$C=\sum_{k=1}^n\cos (k)$$ $$C+i S=\sum_{i=1}^n e^{ik}=\frac{e^i \left(1-e^{i n}\right)}{1-e^i}$$ So, $$C=\Re\left(\frac{e^i \left(1-e^{i n}\right)}{1-e^i}\right)=\frac{1}{2} \left(\cos (n)+\cot \left(\frac{1}{2}\right) \sin (n)-1\right)$$ $$S=\Im\left(\frac{e^i \left(1-e^{i n}\right)}{1-e^i}\right)=\frac{1}{2} \left(\sin (n)-\cot \left(\frac{1}{2}\right) \cos (n)+\cot
   \left(\frac{1}{2}\right)\right)$$
