1. **State the problem:** A particle moves on a unit circle centered at the origin, starting at point $(1,0)$. It undergoes a rotation corresponding to a shape with rotational symmetry order 4. We need to find the smallest angle of rotation in degrees and the new coordinate of the particle after one counterclockwise rotation. 2. **Formula and rules:** The order of rotational symmetry $n$ means the shape looks the same after a rotation of $$\frac{360^\circ}{n}$$ degrees. 3. **Calculate the smallest angle of rotation:** For $n=4$, the smallest angle is $$\frac{360^\circ}{4} = 90^\circ$$. 4. **Find the new coordinate after rotation:** The particle starts at $(1,0)$ on the unit circle. Rotating counterclockwise by $90^\circ$ moves the point to $(\cos 90^\circ, \sin 90^\circ)$. 5. **Evaluate trigonometric values:** $$\cos 90^\circ = 0, \quad \sin 90^\circ = 1$$. 6. **Final coordinate:** The new coordinate is $(0,1)$. **Answer:** The smallest angle of rotation is $90^\circ$ and the new coordinate point after one rotation is $(0,1)$.