Simplify the expression $\sqrt{(3-\pi)^2}$. Then, consider the condition $\sin(3-\pi) < 0$.

1. Problem Description

Simplify the expression $\sqrt{(3-\pi)^2}$. Then, consider the condition $\sin(3-\pi) < 0$.

2. Solution Steps

We are asked to simplify $\sqrt{(3-\pi)^2}$. We know that $\sqrt{x^2} = |x|$. Therefore,

$$\sqrt{(3-\pi)^2} = |3-\pi|$$

Since $\pi \approx 3.14159$, we know that $3-\pi < 0$. Thus, $|3-\pi| = -(3-\pi) = \pi-3$.

Also, we are given that $\sin(3-\pi) < 0$. Since $3 - \pi$ is a negative number and $\pi \approx 3.14$, $3-\pi \approx -0.14$. Since $\sin(x)$ is negative when $x$ is negative, this inequality is satisfied.

Thus, $\sqrt{(3-\pi)^2} = |3-\pi| = \pi-3$.

3. Final Answer

$\pi - 3$