30.7. Find the number of real roots of the equation
$$
4 \cos (2007 a)=2007 a \text {. }
$$
30.7. three
Let $x=2007 a$. Then the given equation becomes $4 \cos x=x$. The graphs of the equations $y=4 \cos x$ and $y=x$ intersect at three points. Thus, the equation $4 \cos x=x$ has three roots. Consequently, the equation $4 \cos (2007 a)=2007 a$ also has three roots.