I am trying to simplify the expression

$\displaystyle \cos\frac{35\pi}{4}+i\sin\frac{35\pi}{4}$

The cosine and sine functions can be reduced modulo $\displaystyle 2\pi$, right? So I wrote this as

$\displaystyle \cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}$

after subtracting $\displaystyle \frac{32\pi}{4}=8\pi$ from each of the arguments. But checking this equality with my calculator I get something different. What am I doing wrong?