We use radian notation. Every rational multiple of $\pi$ has trigonometric functions that can be expressed using the ordinary arithmetic operations, plus $n$-th roots for suitable $n$.
This is almost immediate if we use complex numbers, since $(\cos(2\pi/n)+i\sin(2\pi/n)^n=1$.
But it is known, for example, that there is no expression for $\sin(\pi/9)$ that starts from the integers, and uses only the ordinary operations of arithmetic and roots in which every component is real.
The following more restricted problem has a long history because of its close connection with the problem of which angles are constructible by straightedge and compass.
Let $\theta=\frac{m}{n}\pi$, where $m$ and $n$ are relatively prime. Restrict our algebraic operations to the ordinary operations of arithmetic, plus square roots only, The trigonometric functions of $\theta$ are so expressible iff $n$ has the form $n=2^k p_1p_2\cdots p_s,$ where the $p_i$ are distinct Fermat primes.
A Fermat prime is a prime of the form $2^{\left(2^t\right)}+1$. There are only five Fermat primes known: $3$, $5$, $17$, $257$, and $65537$. It is not known whether or not there are more than five.