1. The problem is to solve the equation $\sin x = x$. 2. We want to find values of $x$ where the sine of $x$ equals $x$ itself. 3. Recall that $\sin x$ is a function that oscillates between $-1$ and $1$, while $x$ is a straight line. 4. For large $|x|$, $x$ will be greater than 1 or less than -1, but $\sin x$ stays between -1 and 1, so no solutions exist there. 5. At $x=0$, $\sin 0 = 0$, so $x=0$ is a solution. 6. For $x \neq 0$, consider the function $f(x) = \sin x - x$. 7. We check the derivative $f'(x) = \cos x - 1$. 8. Since $\cos x - 1 \leq 0$ for all $x$, $f(x)$ is non-increasing. 9. Because $f(0) = 0$ and $f(x)$ decreases away from zero, no other solutions exist. 10. Therefore, the only solution to $\sin x = x$ is $x=0$. **Final answer:** $x=0$