8. Find the number of real solutions to the equation
$$
\frac{x}{100}=\sin x,
$$
where $x$ is measured in radians.
(A) 30
(B) 32
(C) 62
(D) 63
(E) 64
8. Answer: (D)
Since $-1 \leq \sin x \leq 1$, we have $-100 \leq x \leq 100$. We also observe that $31 \pi<100<$ $32 \pi$.
For each integer $k$ with $1 \leq k \leq 16, \frac{x}{100}=\sin x$ has exactly two solutions in $[2(k-1) \pi(2 k-1) \pi]$, but it has no solutions in $((2 k-1) \pi 2 k \pi)$. Thus this equation has exactly 32 non-negative real solutions, i.e. $x=0$ and exactly 31 positive real solutions. Then it also has exactly 31 negative real solutions, giving a total of 63 .