1. The problem is to solve the equation $\sin x = 0.45$ for $x$. 2. Recall that the sine function gives the ratio of the opposite side to the hypotenuse in a right triangle, and its values range between $-1$ and $1$. 3. To find $x$, we use the inverse sine function (arcsin): $$x = \arcsin(0.45)$$ 4. Calculate the principal value: $$x \approx 0.466 \, \text{radians}$$ 5. Since sine is positive in the first and second quadrants, the general solutions are: $$x = 0.466 + 2k\pi \quad \text{and} \quad x = \pi - 0.466 + 2k\pi$$ where $k$ is any integer. 6. Simplify the second solution: $$x = 2.676 + 2k\pi$$ 7. Therefore, the complete solution set is: $$x = 0.466 + 2k\pi \quad \text{or} \quad x = 2.676 + 2k\pi, \quad k \in \mathbb{Z}$$