Find two unit vectors that make an angle of 60 degrees with v = <3,4>.
If $\displaystyle \left\langle {x,y} \right\rangle $ is the vector you are looking for then you know that each of these must be true:
$\displaystyle x^2 + y^2 = 1$ and $\displaystyle \cos (60^ \circ ) = \frac{{\left\langle {x,y} \right\rangle \cdot \left\langle {3,4} \right\rangle }}{{\left\| {\left\langle {x,y} \right\rangle } \right\|\left\| {\left\langle {3,4} \right\rangle } \right\|}}$.
Now solve!